THERMODYNAMICS FOR ENGINEERS CAMBRIDGE UNIVERSITY PRESS C F CLA1, MAN unu LONDON FETTEELANE.E C I N L \\ \ ( ) R Iv. ( P l> LI T N V M ' s M ) N - BOMB VY CA.LCUTT\. M VLMILL VN VN1)( i), [ ii, TOfioNTo J M DFNT \NDsoNs, IIP TDK'S 1 1 MVRU/LN K vnUS^HIKJ Iv \ISll \ THEKMODYNAMICS FOE ENGINEEBS BY J. A. EWIJSTG, K.C.B., M A , LL.D., D.So., F.R.S., M tar C E , M.INST |MECH.E. Pimcipal and Yice-Chancelloi of the University of Edmbuigh , Honoiaiy Fellow of King's College, Cam budge, Foimeilj Piofeasoi of Mechanism and Applied Mechanic^ in the Univcisit) of Cambndge, Sometime Dneotoi of Naval Education CAMBRIDGE AT THE UNIVERSITY PRESS 1920 PKEFACE ALTHOUGH written pnmarily for engineers, it is hoped that this book may be of seivice to students of physics and others who wish to acqiuie a working knowledge of elemental y thermo- dynamics liom the physical standpoint. In piesentmg the fundamental notions of thermodynamics, the wnter has adopted a method which his experience as a teacher encomages him to think useful The notions aie fiist introduced in a non-mathematical foim, the readei is made familiar with them as physical leahties, and learns to apply them to piactical problems, then, and not till then, he studies the mathematical iclations between them. This method appeals to have two advantages it presents the non-mathematical student fiom becoming beuildeied on the thieshold, and it saves the mathematical student fiom am iibk of failing to realize the meaning of the symbols with which he plays When the non-mathematical student comes to face the mathematical relations, which he must do if he is to pass beyond the indiments of the subject, he finds it compaiatn ely easy to build on the foundation of physical concepts he has aheady laid thcie is perhaps no bcttei \\ay to leain the meaning and u^e of paitial diffeiential coefficients than by applying them to tliermo- dynamic ideas, once these ideas aie cleaily appiehended. Accoidingly the plan of the book is to begin with the elemental y notions and then mteipretation in piactice, and to defei the study of gcncial theimodynamic lelations till neai the 2nd Finally these relations aie illustiated by applying them ,o charactcnstic equations of fluids,, and in paiticulai to steam, "olio wing Callendai's method The chaplei on Inteinal Combustion Engines gives occasion or intioducmg some icsults of cxpeiiments on the internal 'iiergy and specific heats of gases, and this matter is dealt vith fmther m an appendix which attempts an elementary iccount of the molecular theory. In any exposition of the first principles of theimodynamics t is important to choose a way of dealing with temperature uch that students may be led by simple and logical steps to nderstand the thermodynamic scale. The course followed vi PREFACE here ib first to imagine an ideal gas which serves as theimometnc substance, and also as the woikmg substance in a Carnot engine. This gives a perfect-gas scale by inference to A\hich the efficiency of any Carnot cycle is provisionally expressed, and from that the step to the thcimodynamic scale is easy The writei is indebted to Piofessoi Callendai and his publisher, Mi Edward Arnold, lor permission to include a much abbieviated veision of his Steam Tables. By the lecent publication of complete Tables, Piofessor Callendar has added substantial!)' to the many obligations imdei \\lncli he has put all students of thermodynamics. The writer would also thank Mi J. B Peace, of the Cambiiclge Umveisitv Press, foi vanous suggestions and foi the mteiest lie has taken in bunging out the book, and also Di E. M Hoisburgh, of the Mathematical Depaitment of this Univeisity, for his gieat kindness in reading the proofs. THE UNIVDRSI-H, EDINBURGH MaiJi 1920 CONTENTS - CHAPTER I < . FIRST PRINCIPLES The Science of Thermodynamics Heat-Engine and Heat-Pump Efficiency of a Heat-Engine Coefficient of Peifoimance of a Refrrgeratmg Machine Woilang Substance Opeiation ot the Working Substance in a Heat-Engine Cycle of Operations of the Woilang Substance Thp Fust Law of Theimodynamics Internal Eneigv Woik done in Changes of Volume of a Fluid Indicatoi Diagiams Unit*, of Foict 1 , Pievsine, and Woik Units of Heat Mechanical Eqrmalent of Heat Scales of Tcmpeiatuie Reckoning of Tcmpeiatiue fiom the 'Absolute Zeio" Piopeities ot Gases Chailes' Lan and Bo\lf\ Law Notion of a 'Perfect" Gas Inter iiti! Eneigv of a Gas Joule's Law Specific Heats of a Gas Constancy ot the ^pccihc Heatb in a Peifect Gat, Reveisible Actions Adiabatic Expansion Isothermal Expansion Adiabatic Expansion of a Peitt ct Gas Change- of Tcmpeiatuie in the Adiabatic Expansion of a Peifect Gas Woik done in the Adiabatic Expansion of a Peitect Gas Isotheimal Expansion of a Peifect Gas Summary ot icsults foi a Perfect Gas Fundamental Questions of Heat-Engine Efficiency The Second Law of Theimodynamics Reversible Heat Engine Cainot's Cycle of Operations Carnot's Principle Reversrhilrty the Criterion of Petfection in a Heat-Engine Efficiency of a Reversible Heat-Engine Cainot's Cycle wrth a Peifect Gas for Working Substance Reversal of this Cycle Efficiency of Any Reveisible Engine Summary of the Argument PAGE 1 1 2 2 2 3 4 5 5 (3 1 8 9 10 10 12 13 14 15 n 19 20 21 22 22 23 24 24 25 26 26 27 28 29 29 30 32 34 34 AST 40 Absolute Zeio of Tempeiatme 35 41 Conditions of Maximum Elh&iency 3G 42 Theimodynaniic Scale of Tempeiatme 39 43 Revcisible Engine lecemng Heat at Vaiioiva TompPiatiups i2 44 Enhopv . 44 45 Conseivatiou of Entiopv in Ca, mot's Cycle 45 40 Ent;opy-Tcmpeiatuie Diagiam foi Cainot's Cycle 46 47 Enfcropy-Tompeiatiue Diagianit, ioi a sonca of TCeveisiblc Engines 47 48 JSTo change of Entiopy in Adiabatic Pioceasos 48 49 Change of Entiopy in fin Iuo\ciaiblo Opeuition 48 50 Sum oi the Entiopiea m a System 49 51 Entiopy-Tempeiatuie Diagiams 50 52 Peifect Engine using Rogoneiatoi 52 53 Staling' a Regonpiative An Engine 53 54 Joule's Au-Entrino 55 CHAPTKR II PROPERTIES OF FLUIDb 55 Slates of Aggregation 59 56 Foiuiafcion of Steam undei Constant I J jcs=;iue GO 57 iSdtuifUod and Supci healed Steam 01 58 Relation of Piessuie to Tempeiatiuo in Satiii.itcd SI earn 02 59 Tables of the Piopeities of titeam 62 00 Relation of Piessiue to Volume in Sat aid ted Steam 0-1 61 Boiling and Evapoiation 05 G2 Mi\tiuo ol Vapoiu with othei Gases Daltou's Pniui|jU l 05 03 Evapoiation into a space contciiinng An Hatinadon of the Atmospheic with Watci-Vapoiu 60 01 Heat ipqiiued foi the Poi (nation, ot Steam undi'i C'onstant PHS^UIC- Heat of the Liquid and Latent Hcvifc 07 05 Total E\teinal \Voik done 09 06 Internal Eueigy of a Fluid 09 07 The Total Heat" of n Fluid 70 OS Change of the Tola! Heat dining LTenhng undei C'onst.mt Piessmcs 71 69 Application to Steam foniu-d undei Constant Piessuie, horn WatuafcOC 71 70 Tot\l Heat of a mixliuo ol Liquid and its Satiualed Vapom 73 71 Total Heat of Snpei heated Vapom . 73 72 Constancy of the Total Heat in a Tin ott ling Piocoss 74 73 Entiopy of a PI n id 75 74 Mi'sied Liquid and Vapom \Y<-( Steam 70 75 Specification oi the Slato of am Flmd 77 70 Isotheimal Expansion of a Fluid Jsofheinml Lmrs on (lie Piessuie-Volumo Diagiaui 78 77 The Ciitical Point CuticaJ Tcmporadue and Cutical Piessmo 80 CONTENTS IX ART PAGE 78 Adiabatic Expansion of a Fluid . 8J 79 Supersatmation 84 80 GWngc of Internal Energy and of Total Heat in Adiabatic Expansion "Heat-Drop" 86 CHAPTER III TIIEORV OF THE STEAM-ENGINE 81 CdinoL'a Cycle with Steam or othei Vapoui foi Woilang Substance 88 82 Efficiency ot a Perfect Steam-Engine Limits of Temperature 90 83 Euttopy-Tomperature Diagram for a Perfect Steam-Engine 91 Si [Jso of "Boundary Curves' in the Entropy-Temperature Diagram 92 8'") Modified Cycle omitting Adrabatie Compression 94 8f> Engine with Separate Organs 96 87 The RanLme Cycle . 98 88 Eihciuicy of A Rankme C^clo 99 8 ( ) Calculation of the Heat-Drop 100 ')() The 1 Function (,' 102 Ml Intension ot t!ie Rankine C\cle to Steam supplied in any State 104 M2 Ro.nk.iiK Cycle with Steam initially Wet . 104 9-} Rankine C\ do ml h Steam nutiall\ Snpeilicatcd lOb <n l{e\c-isibihtA of the Ilankinc CVIe 10 l ) M, r > Conditions ot High Efficiency 110 Ml> Elite ( of Incomplete Evpan^ion 112 M7 Idral Engine u 01 king \uth No Expansion 114 MS Clapn ion's Equation ll r i MM \])plu.ition oi Cla,[x\ ion's Equation to utliei Changes in IMnsual Mali- lib FOiiliop^-TempLi.iluie C'hait of (he 1'iopeities ot Steam 118 Molliei's (.'hail ot Entiopy and Total Hi.it 121 Othoi Foi ms of Chart 125 blik'cts of Tlnottling 120 The Heat- Account in a Real Pioccss 129 CHAPTER IV THEORY OF REFRIGERATION 'fi Tho liofngei.ition Process 133 (! KovciHiblo Kofiigerating Machine 134 7 Conaeivation of Entrop3 r in a Perfect Refrigerating Piocess 135 8 Ideal Coofhcionts of Performance 136 [) Tlio Working Fluid in a Refngeiatmg Pioccss 137 1 Tho Actual Cycle of a Vapour-Compression Refngeiating Machine 138 1 Entropy-Tempeiature Diagiam foi the Vapoui -Compression' Cycle . 141 x CONTENTS ART rvon 112 Rcfngciatmg Enect and Woik of Completion, exploded 111 Tcims of the Total Heat 144 113 Cho,its of Total Heat and Entiopy foi Substanceb u^cd 111 the Vapoiu-Compiession Piocess 145 114 Applications of the Iff) Chait in studying the Vaponi Compres- sion PiooebS 149 115 Vapoui'-Compiession by means of .1 Jet W.itoi -Vapoui Machine 1 55 110 The Step-down in Tcmpeiatiue Use of an Expansion Cylinclci in Machines using ALL 157 117 An -Machines Joule's An -Engine leveised l r >8 118 Dnect Application of Heat to pioduce Cold Absoiption Machines 30(1 119 Limit of Ellicieney in the Use of High-tempeiatuic Heat to PiodnccCokr 104 120 Expiession in Tei 111-5 ot tlie Entiop\ !()(> 121 The Rcfiigciating Machine at, a means of \Vainniig l(>8 122 The Attainment of Veiy Low Tempeialuie Cascade Method 1(>9 123 Regeneiative Method 171 124 Fust Stage 172 125 Second Stage 175 126 Lmde's Appaiatus J7to 127 Liquefaction of An by Expansion in which Woik jt> done Claude's Appaiatus 178 128 Sepaialion of the Constituents of An 180 129 Ba^'sCmves 185 1,30 Complete Rectification 188 CHAPTER V JETS AND TURBINES 131 Theoiy ot Jets 191 132 Fonn of the Jet (De La\ al's No/7le) 103 1M3 Limitation ot the Disehaige thiough an Oiifice of Cuen Size 197 134 Application to An 198 135 Application to Steam 199 130 C'ompaiison of Mctastablc Expansion i\ith Equihbiium Ex- pansion 20'j 137 Measuie of Supcisatuiation 20(> 138 Rctaidcd Condensation 207 139 Action of Steam in a Noz'lc, continued 208 140 Eltects of Faction 20 ( ) 141 Apphcaiion to Tuibineg 214 142 Simple Tin bines 2I r > 143 Compound Tmbine's 21 b 144 Theoretical Efficiency Ratio 21(> 145 Action in Successu e Stages 217 140 Stage Efficiency and Reheat Factoi 218 CONTENTS xi ART PAGE 147 Ecal Efficiency- Ratio . 219 148 Types of Tin bines . 220 140 Peifoimancc of a Steam Tmbme 222 150 Utilization of Low Piessme Steam 223 CHAPTER VI INTERNAL-COMBUSTION ENGINES 151 ' Internal Combushon . 225 152 The Foui-Stioke Cycle . 226 153 The Cleik 01 Two-Stioke Cycle . 226 15 J- Ideal Action 227 155 AH Standard 229 15G Constant-Pie&binc Type 232 157 Diesel Engine 234 J5S Combustion ot Gases Moleculai Weights and Volumes 235 151) The Giammo Molecule 01 Mol 237 IbO The Um\eisal Gas Constant 238 Ibl Specific Heats of Gases in Relation (o then Moletulai Weights VoIumcUK! Specihe Heats 239 Ib2 Suminaiy ot Methods oi evpiessmg the Specific Heats 241 1(53 Measui eel Values ot Specific Heats 241 101 Vaiialion nt Specific Heat -\\ith Tempeiatuie 243 11)5 TiUcinal Eneigy ot a Gas 245 l(i(> Aehabatic Expansion of a Gas \\ith Vanable Specihe Heat 247 l(>7 Ide.il Elhciency as aflccted by the Vanation ot the Specific Hc.il \\itliTeiupeialine 241) 1()S ( 'line i)t liitunal KnugN toi Topical Gas Engine Mixtuic 251 Hi!) Action in a Heal Engine Anal\si-, ot the Inclieatoi Diagiam 254 J70 Me asuie inent ut Suction Tempeiatuie. 257 171 The Piocess ot Explosion 257 172 F fleet ot Tinbulvnce 259 173 Radiation in E \plosions 2bO 171 Moleculai Eneigy of a Gas 2bl 175 JJisaocialion 2L>5 CHAPTER VII GENERAL TIIERMODYNAMIC RELATIONS 170 Inliodiictiou 200 177 Functions ot Iho State oi a Fluid 206 178 Relation of any one Function of the State to two othci & 267 179 Encigy Equations and Relations deduced fiom them 270 180 Expiessions foi the Specific Heats K u and Jv ;) 272 181 Fuithcu deductions fiom the Ecaiationa foi E and 1 275 182 The Joule-Thomson Eucct . 276 183. Umesisted Expansion . 279 X11 CONTENTS ART TVGE 184 vSlopes of Lines m the 70, T$, and IP cliaita, foi any Fluid 280 185 Application to a Jlixtuie of Liquid and Vapom in Equilibi mm Clapeyion's Equation Change of Phase 288 186 Compiessibihtv and Elasticity of a JFlnicl 281 > 187 Collected Results 2S(> CHAPTER VIII APPLICATIONS TO PARTICULAR FLUIDS 188 Ch.uActeustic Equation 2J)0 189 Chaiactenstic Equation of a Pcifect Gas 290 190 IsotliPimal and Adiabatic Expansion oi Ideal Gat, 2!)2 191 Entiopy, Eneigy, and Total Heat of Ideal Gas 20 > 192 Ratio of Specific Heat-. Method o infemnj; y in Gasi.'a iom the Obseived Velocity ot Sound 203 193 Mea-smement of y bv Adiabatic Expansion Mt'thod ot C'16- ineut and Desoi ines 204 194 Eftecfc of Impei Section of the Gab on the Ratio of Specific Heats 20fi 195 Relation of the Coohng Effects, to the Coefficients of Expansion 20b 196 Foims of Idotheimals Diagiams oi P and V, and of PV and P 208 197 Iinpeiiect Gat.es Amngat\ Isothennals of PV and P 20 ( ) 198 IbOtheimals on the Picssuie-Volume Diagiain JOo 199 Continuity ot Liquid and Gas J04 200 Van clei Waals' Chaiacten^tic Eqmtion .{()<> 201 Cutical Point accoiding to Van dti WaaU 1 Equation .'{O'J 2(J2 Coiiesponding States ;j|j[ 203 Van clei Waals 1 Equation only Appio\nnate 204 Othei Chaiacteiibtic Equations, Claubiu-,, Lheteiici 205 C'allendai's Equation ;jiy 206 Deductions fiom the Callcndai Equation J21 2U7 The Specific Heats, in C'allendai'b Equation J24 208 The Entiopy, Eneigj , and Total Heat, m Callcnd.u\ ]^|uation ,'J2, r > 209 Apphcation to Steam ^27 210 Total Heat and Entiop\ ot Watei ;jjj 211 Relation oi PiCbSine to Tempeiatuie in Satiuatecl Steam ).!( 212 Foimukb ioi the Latent Heat ot Steam, and foi thr \ r olumc of a Wet Mi\tuie jjg 213 Collected Foimulas. toi Steam yjg 214 Tables ot the Piopei ties of tsteam jjjO APPENDIX I EFFECTS OF SURFACE TENSION ON CONDENSATION ^ND EBULLITION 215 Natuie ot Suiface Tension 342 216 Need of a Nucleus gj7 217 Kelvin's Pimciple . *,*> 218 Ebullition CONTENTS 219 220 22] 222 223 224 APPENDIX II MOLECULAR THEORY OF GASES PICHSIUG clue to MolecularJ'.aijja'ils Boyle's, Avogadro's, and Dalton's La\\ s Pei feet and Impeifecfc Gases Calculation of the Velocity of Mean Squaie Internal Eneigy and Specific Heat Energy of Vibiation PJanck's Formula PAQE 351 355 350 350 357 361 3(>2 220 Effect of Extieme Cold on the Diatomic Molecules of Hydiogen 360' APPENDIX III TABLES OF THE PROPERTIES OF STEAM \B1 L \ Piopeities of Satin ated Steam, in lelation to the Tenipeiatine 308 A Propci ties of Watei at Satmahon Pressine 304 B L'topeities ot Saturated Steam, in relation to the Piessme ^70 ( ' Volume ot Steam in any Diy State 372 I) Total Heat ot Steam in any Di\ State 374 10 Entiopv ol Steam in any Di> State 370 I-' Specific Heat, at constant pics^uie, ot Me. am in an\ Di\ i^tato 378 IN DUX . . . 37'J CHAPTER I FIRST PRINCIPLES 1 The Science of Thermodynamics ticals of the relation of at to mechanical woik. In its engmeeimg aspect it is chiefly con- incd with the pioccss of getting woik clone thiough the agency of at Any machine i'oi doing this is called a Hcat-Engme. It is *a concerned with Lhc pioccss ol'icmoMiig heatfiom bodies thai" e cilieady coldei than their smioundings Any machine foi doing is is called a Refngciating Machine It is coinement to study the theimod> namic action of heat- i>incs and ichigciating machines togclhei, because one is the v cise of the olhi'i, and bv consideiing both \\e aimc moie easily an understanding ol I he whole subject. 2 Heat-Engine and Heat-Pump In a f Icat-Enginc heat is pplu-d, geneially bv Ihe combustion ot luel, at a high tempcia- ic, mid the engine dischaigcs heat at a lo\\ci tempciatuic Thus a sleam-eni>ine heal is taken in at the tempciatuic ol the boilci id chsc'haigcd at Llie tcmpeialnic ol the condensei In any kind luMl-cni>mc Ihe heal is lei do\ui, \\itlim the engine, honi a high vcl ol leinpe'iatiiu' lo a. lowci lc\d ol Icmpciatme, and it it, by III ling heal down lh.it Ihe engine is, able to do woik, as a ilci-whccl is able to do woik b} letting \\atei doun fioni a high \cl to a lowei level Bui theie is this mipoitant dillcience, that me ol the heal disappcais in the pjoeess ol being let clown it is 'liveried into the woik which the engine does In a Refrigerating Machine woik has to be spent upon the aclinic to enable il to lake in heat at a, low level of temperatmc, id dischaigc heat at a highci level of tempeiatuie, just as work Duld have to be spent upon a wa Lei -wheel if it were used as a eanb of laising watei by levcismg its action, in such a way that e buckets wcie filled at a low level and emptied at a higher level, lhat it tJiould seive as a, pump. It would be cjmte coricct to cak of a refuge-rating machine as a heat-pump. But again theie an important difference between the refrigerating machine and E T i 2 THERMODYNAMICS [en. the r ever scdAvater- wheel the refrigerating machine is a heat-pump which dischaiges moie heat than it takes in, foi the work Avluch rs spent m driving the machine is converted into heat, Avlnch has to be dischaiyed at the higher le\ el of tempeiatuie in addition to the heat that is taken in at the IOAV tempeiatuie. 3. Efficiency of a Heat-Engine Fiom the point of VIBAV of piactical theimod^namics the object of a heat-engine is to get Avork done with the least possible expenditure ot fuel In other woids the ratio of the A\ork clone to the heat taken in should be as laigc as is piacticable This ratio is called the Efficiency of the engine as a hcat-engmc The theory of heat-engines deals \\ith the conditions that alfect efficiency, and Autli the limit of efficiency that can be i cached when the conditions aie most favourable 4. Coefficient of Performance of a Refrigerating Machine In a refngeiating machine Lhe object is to get heat icmovcd liom the cold bod}'' and pumped up to a lughei lc\ el oi tempeiatuie at which it can be discharged, and what is wanted is that this should be done with the least possible expenditure ot woik. The latio of Lhe heat taken in by the machine from the cold boel} to the woik that is spent in driving the machine is called the Cocllicient of Peifoim- ancc. The theory oi lefiigeiaLion deals with the conditions that AVill allow tint, latio to be as laige as possible 5 Working Substance. In the action of a heat-engine or oi a leiiigeiatmg machine there is ah\ r a\s a woiking substance which foims the vehicle by which heat passes through the mac-hinc It is because the woiking substance has a capacity ioi taking in heat that it can art as a \chiclc for coin eying heat horn one level oi tempeiatuie to another In this process its volume changes, and it it, by means of changes of volume on Lhe pait of Lhe working sub- stance thai Lhe machine docs woik, it it is aheal-cngmc, 01 has work spent upon it, if it is a icfngeiating machine. Accordingly, an important part of the science of thermodynamics deals with the > piopcities of substances m i elation to heat, and the connection between such piopcities m any substance. The substances with which AVC aie chiefly concerned aie fluids in the gaseous or liquid i states They include an and other gases, \vater and Avater-vapour, J and also some fluids moie easily vaponzed than Avater, such as \ ammonia and carbonic acid, which aie used as the workmo sub- I * O f stance in ceitam lefng era ting machines. Each fluid has of course i] FIRST PRINCIPLES 3 its own charactenstics ; but many of the relations between its pro- peilies are of a general kind and may be studied without limitation to individual fluids It will be seen, as we go on, that much of what has to be said applies equally, whatevei fluid selves for woikmg substance, and that in any one fluid the various pioperties are connected with one another in a -way that is true for all fluids. The study of the theimodynamic lelatioiibhips between the vanous piopeiLics of a fluid is useful, not only because of the dnect light it llnows on the action of heat-engines, but also because it enables a practically complete knowledge of the piopeities of a fluid m detail to be inferred fiom a compaiatn ely small number of expeii- menlal data We bhall see latci, for example, how such iclation- ships have been made use of in calculating modem tables of the piopeities ol steam fiom the icsults of careful measuiements, made in the laboiatoiy, of a \eiy few fundamental quantities. 6 Operation of the Working Substance in a Heat-Engine. In oencial the uoikmg substance is a fluid which opeiates by chang- ing its \olumc, cxeiling piessiue as it docs so But it is easy to imagine a hcat-cnginc ha\uig a solid boch foi uniting substance, say a. long loci ol metal anangcd to act as the pa\\l of a latchet- whc'el wilh closely pitched teeth Lei the lod be heated so that it lengthens sulUciuUl\ to dine the \\heel foi \\aid thiough the space ol one tooth Then let the iod be cooled, sa\ bvapph ing cold water, llic iatchct-\\hccl being meanwhile held liom letuining by a separate click or detent The iod on cooling uill letiact so as to engage itself uith the next succeeding tooth, uhich may then be eh i ven foi \\ aid by heating the iod again and so on To make it evident that such an engine would elo woik \\e have onl) to suppose thai the latchct- wheel canies lound with it a chum by which a weight is wound up The device forms a complete heat-engine in which the woikmg substance is a solid iod, doing work m this case not through changes of \olume but thiough changes of length. While its length is mci easing it is exerting foice m the dnection of its length, ft receiver heat by being bi ought into contact with borne souice of heat at a compaiatively high tempeiatme, it tians- foims a small pail of this heat into woik, and it rejects the re- mainder to what we may call a icceiver of heat, which is kept at a compaiatively low tempeiatme. The gieatei pait of the heat may be said simply to pass through the engine, from the souice to the reccivei, becoming degraded as legaids temperature in the process 12 4 THERMODYNAMICS [en. This is typical of the action of all heat-engines, they con veil some heat into woik only by letting down a much largci quantity of heat fiom A high tempciatuic to a iclativcly lo\\ tenipeiaLurc. The engine we have just imagined would not be at all effiacnl, the fi action of the heat supplied to it which it could convcit into woik would IxMciysmall Much gicatci efficiency can be obtained by using a fluid Joi woiking subbtancc and by making it acl bo I hat its own expansion of volume not only docs woik bnl also can-.es it to fall m tcmpeiatiue bctoie it begins to lejcct heat Lo the cold icccivei. 7. Cycle of Operations of the Working Substance. Ccncially in the action ot a heat-engine or ol a lelnaeiaLing machine the woiking substance ictmns periodically to the s.mic state of tcm- pcialuie, picssmc, volume and physical condition m all lespecls Each Lime this has occuncd the snbslancc is said to have passed thiough a complete cycle of opciations lun example, in a con- densing steam-engine, watci taken fiom the hoi-well is pumped into the boilci it then passes into the cyhnelu as si earn then liom the cyhndei mlo the condcnsei, and finally fiom the condense i back lo the hot-well, it completes I he cycle by ulunimn lo the same condition m all icspects as at Insl, and is ic.idv lo go Ihiough the cycle again In othei less ob\ ions cases a hi lie considcialion shows that the c\ clc is completed although lhe i same pen I ion oi \voikmg substance 1 elocs not go thiough il a<_>am thus in a non- conelensmg sleani-engme the slcam which has passed thiough the engine is disehaigeel into Hie atmosphcic, wheic it cools to the tcm- pciatuic of the feed-watt. i, while a Ciesh poilmn. ol Lccl-watci is deh\cie'd to Ihc engine to go llnongh the. cycle in its turn In the theory o( hcal-cngmcs it is of the lusl nupoi lance to con- sidei as a whole UK, cycle- of opeiahons pcifoinu'd by I lie 1 woiking substance. If we t>top shoi I of the completion of the cycle ma I leis aie complicated by the fact that 1hc substance is in a slate difh'icnt from its initial state On t he eilhei hand, if the cycle is complete we know that whalevci heat 01 othci energy the substance conlamul within itself I o begin with is tlieic still, for the state of the substance is the same m all respects, and consequently any woik that it has done must ha\c been done at the expense of heal winch il has taken m during the cycle The total amounl of cneigy it has paitcel with must be equal to the amount it lias icceivcd, duimg the cycle, foi its stock of internal eneigy is the same al the end as at the i] FIRST PRINCIPLES 5 beginning We can al once apply I he pimciple of Ihe Conservation of Energy and say that for Llic cyclic process as a whole this equa- tion must hold good, Heat taken in = Heat rejected + Work done by the substance And sinnlaily, when the working substance in a lefngeiatmg machine has been cainecl through a complete cj r cle of opeiations, the equation holds for the c} cle as a whole, Heat taken in = Heat i ejected - Work spent upon the substance. 8 The First Law of Thermodynamics. The pimciple of the Conscuation of Encigy m i elation to heat and work may be ex- pressed m the following statement, which constitutes the First Law of Theimodynamics When mechanical energy 'is produced from licat a definite quantity of heat goes out of existence fo> evety unit of too) L done, and, conveischj, when heat is produced by the expendi- ture of mechanical energy the same definite quantity of heat comes into existence for every unit of waiL spent 9 Internal Energy. We have used in Ait 7 a plnase which icquncs some Imthci explanation the intcnial eneigy of a sub- stance No means exist by which the ^holc stock o eneigy that a substance contains can be measnicd But \\c aie concerned only \\ith changes in that stock, changes which ma} ausc fiom the sub- sLancc taking in 01 gnmg out heat, 01 doing \\oik, 01 haMiig \\ork spent upon it U a substance takes in heat \\ithont doing -\\oik its slock of mil null cncigj mciiases bv an amount equal to the heat taken in II it docs \\oik without lakmg m heat, it docs the woik at the expense ol ils slock oi internal eneigy, and the stock is diminished by an amount equal Lo the woik done In geneial, when heat is being taken in and the substance is at the same time doing woik, we have Heat taken in = "\Voik done + Inciease ot Internal Eneigy. Foi any mfimtcsimally small step m the pioccss, we may wnte dQ = dW + dE, wheic city is the heat taken in dming the step, dW is the woik done, and dE the increase of internal eneigy. In a complete cycle thcie is, at the end, no change of the internal cneigy J3, and consequently foi the cycle as a whole, Qi - Q. - W, where Qi Q.Z 1S ^e net amount of heat received, namely the 6 THERMODYNAMICS [en. difference between the heal taken in and I lie heal i ejected in Lite complete cycle and IF is the woik done in the complete cycle In this notation \\c aic supposing }V to be expressed in units oi' heat, as well as Q and E It would be more eonect to speak oi' W as Hie thcimal equivalent of the woik done. 10. Work done in Changes of Volume of a Fluid In an engine of the usual cyhndci and piston type the working fluid docs woik by changes of \olumc. The amount oi 1 ^ork done de- pends only on the relation of the press me to the \oliimc m lhc.se changes, and not on the form of the vessel 01 vessels in which I lit changes ot volume take place. Let the mtensit t y of prcssmc of Hie fluid (that it, to say the picssme on unit of aica) be P while the piston moves foiwaid through a small distance Sf If the aica of I he piston is S the total foiec on it is PS and the woik done is PM81 But SSI = 8T, the change of \ olume hence the woik done is P&T iV. for the small change of volume 8V, or PdV foi a finite change r, of volume horn a volume T r 1 to a volume V , dining which the prcssme ma} 7 \ ary. In any complete cycle of operations the volume at the finish is the same as at the start, and the woik done is IPclV taken round the cycle a,s aAvholc It is vcrv useful to icpuscnt oiaplncallv I he ^\oik which .1 fluid does in ehaiinmg its w>lumc' Lcl a diagiam be diawn m which the relation of the pics- sme of any sii])poscd woiking substance to its volume is shown by rcctangiilai cooieluuitcs as in fig 1. Beginning with the state jo rcpicscntcd by the point A, wheic f> the pressiuc is AM and volume OM, suppose the substance to expand to a state B, wheie the pressure is UN and the volume M N ON, and let the cm\ e AB rcprc- Volume sent the mtermeehatc states of ^ ^ prehsuie and volume. Then the work clone by the substance in this rON expansion, which is PdV, is represented bv the aiea MABN ] OM under the cuive AB. i] FIRST PRINCIPLES 7 Again, if the substance undergoes any complete cycle of change (fig 2) by expanding A from A through B to C and by being compressed back through D to A, woik is done by it while 1) it is expanding from A ^ to C, equal Lo the aica MABCN, and woik is spent upon it while it is M N being compicssed fiom Volume C thiough D to A, equal Fl 2 to the aiea NCDAM The net amount of woik which the substance does dining the cycle is equal to the algebiaic sum of those aicas in other woids it is equal to the area of the closed figuie ABCDA icprcscnting the complete cyclic opeiation, which aica is PtlV. If on the othci hand the operation were such as to tiace the fig ui c in the opposite diiechon, the substance being expanded fiom A to C tluongh D and compicssed fiom C to A thiough B, the enclosed aica would be a measiue of the \\ork expended upon the substance in the cycle. ii Indicator Diagrams This picssmc-'s olume chaoiani is an example, and a genciali/ation, of the method of icpicscnting woik which Watt iiiliodiiccd by his im ention ot the Indicatoi, an instalment foi automatically di awing a chagiam to icpiescnt the changes of prcssmc in i elation to changes of \olumc in the action of an engine The figuie ABCDA may be called the Indicate) D tag) am of the supposed action. The mdicatoi consists of a small cj hndei containing a piston which can move in it without sensible fiiction but is contiollcd by a stiff spi ing. This is put in free communication with one end of the woiking cylinder of the engine, so that the woikmg substance presses on the indicator piston and displaces it, against the spnng, thiough distances that are proportional to the piessure at eveiy instant. Connected with the mdicatoi piston is a pencil which rises or falls with it, the connection being made, geneially, thiough a lever that gives the movements of the indicator piston a convenient magnification. A sheet of paper on which the pencil maiks its 8 THERMODYNAMICS I car. movements is caused to move throuqh distances piopoilional to the motion of the engine piston, and at light angles 1o Ihc palh oi the pencil Thus a dmgiam is drawn like thai ol 1 Jig. 2, exhibiting a closed cuive loi each double stioke ol the engine piston, .mil wilh cooidmates which icpicsent the changes ot piess.ne and changes of volume The enclosed aiea, when mtcipiclcd l>\ icleicnex to the appropriate scales of piessme and volume, mcasmes I he ml amount of woik done in the engine cyhndei dm ing [he double stioke, so fai as, one side of the piston is conccincd If Ihc engine is double-acting that is to say, if the woikmg snbsl.mcc .u-ls successively on the two sides of the engine pislon dm ing successive- strokes a similar indicator diagiam is taken foi Ihc ollui end of the cyhndei as well. 12 Units of Force, Pressure, and Work Foi engine einig purposes, in speaking of piessme and ot woik, Hie common mill ol force m Bntish and Amcncnn iihagc is the weight ol 1 Ib and in continental usage the weight of 1 kilogiammc 1 J3y I he woid "weioht" we mean hcic Lhe force with which UK eailh iilh.icls O these masses When it is necessaiy to give sciuitidc puvision lo either of these units of foicc one must spccif\ <i locahly, 01 i.illui ,i latitude, because gumly acts lathci mou shongh as we no horn the equatoi towauls the pole. The s.uue piece ol malmal is nmie stiongly atliactcd b) the caith m London th.m m Pans, lo Ihe extent of one pait in 5000, and moic slionply in London lli.in in New Yoikto the extent of one pait m 1000 If llu weiulil >!' 1 II) of matlei m mean latitude (J5) be taken as nnily, ils weight in any othei latitude A is 99735 (1 + 00053 sin 2 A) The differences due to lahhiele aie so small thai foi neatly nil pmposes they ma^ be ignoicd. The usual units of pitssnrc aic the pound per squ.ire me'h and llu kilogiammepei sqnaic ccnlinicticf Anolhei nml often used is the "Atmospheie," M Inch means the prc.ssme ol the almospheie ^vilh the baiomcter standing at 7GO mm. in latitude I. 1 } , 01 ?5!> (> mm m London This is equal lo a piessuic m London e>f 1 I OSO ])onnds pei square inch or 103274 kilogiammes pci sqnmc ccnlimelie Poi scientific purposes the absolute (eg s.) unit of piessme, Ihe * One lalogiEimino is 2 20462 Ibs f Sini e 1 centimetre is 393702 inch, 1 kilogramme pm iq rm is 14 22D jtoinulH per sq^ in , ^lien both aio measuied at the sauao place, &o Uiat gmviLy uci^ aliku on the pound and the kilogramme - L (\ i] FIRST PRINCIPLES ' , 9 dyne per sq. cm., has the advantage thai it is independent," of; gravity. One " Atmospheie" is equal to 1'0133 x ]0 dynes peY M sq. cm., at any place. Press in es are also sometimes given in inches, 01 in millimeties, of mercuiy. One inch of meicury (at C ) is equivalent to 4912 pounds pci square inch, one millunetie of meicuiy to 1 3596 giammcs pci sq. oni. The usual engineering units of woik are the foot-pound and the mclre-kilogramme 01 kilogiammctre. One kilogiammetie is 7 233 loot-pounds 13. Units of Heat. For the purpose of leckonmg quantities of heat we compaie them with the quantity that is icquiied to warm a unit mass of watci fiom the tempcratme of melting ice to the tcmpciature at which water boils nuclei a piessuie of one atmo- spheie These L\\ r o points sen e to dcteimme two fixed states of tcmpciatuic that me quilc definite and aie independent of the pailiculai way in which lempeiatuie may be measmed The unit ol heal which i 1 - obtained by taking a ceitain iiaction of this riuantilyofhc.il i^ dcscnbcd as Ilie mean thermal unit The mean Iheiiual unit which will be used hcie is one-hunchedth pait ot the heat lequucd to waim one pound of \\atei lioin the melting point to Ihc boiling point at a piessuie ul one atmospheic Tins unit is called Ihe Pound-Cnlon The icason why one-hundiedth pait is lakcn is lliat on tlic Ccntigiadc scale of tempeiatmc the mteival b( Lwccn these (Kcd points is cli\ idcd into 100 degrees consequent!} the pound-calm > is the nvaugc amount ot heat icquiied to waim a pound of walci Ihiouyh one clcgice Centigiade, between the melting poml ,md Ihc boiling point as limits The actual amount lequued pci device uctd not be I he same loi each clegicc of the scale, and in fact is nob the same, for the specific heat ol water is not quite constant. Si milai-ly, what is commonly called the BiitishTheimalUmt (when the Fn.lnenh.cil scale is employed) would be defined as 1/180 of Ihc quantity of hcab required to waim lib of watei fiom the melting point to the boiling point, because on the Fahienheit scale there are 180 degrees between the two fixed points Again, the " Kilo-Caloiy " is one-hundredth of the amount of heat rcquucd to waim 1 kilogiamme of watei from the melting point to the boiling point, and the "gramme-caloiy " is one-thousandth of a kilo-calory 10 THERMODYNAMICS [en. 14. Mechanical Equivalent of Heat. The experiments of Joule, begun in 1843 and continued for scvcial ycais, demonstrated that when work is expended m producing heal a definite rcl.ihon holds between the amount of hcab pioducccl and Llic amounl of woik spent Causing the potential cneigy of a nuscd weight lo he- used up in tinning a paddle which generated heat b\ slnimg water in a vessel, and obseiving the use oi Lempcratuic so ])roducccl 5 . Joule made the fiist determination of the numbci ol units of woik I ha I aie spent in producing a mub of heat This number is called I he mechanical equivalent of heat. Joule found thai 77'2 fool-pounds weie reqmied to raise the tempeiatuie of one pound of walcr thiough one degiee (Fahicnheit) on the LhcimomcLci he employed, at a paiticulai part of the scale Many later and moie exact dcteiminnlions wcic made by Jonh- himself and by other obberveis, using vaiions mclhods of cxpui- ment They agree m shoeing thai Joule's oiigmal figuu' \\as lathei low. The general result is to fix 1400 as Lhc number ol lool- pounds (in the latitude of London) that are equivalent, lo oiu Pouncl-Caloiy as defined in Ait. 13 The coiicspondmg value of the mechanical equivalent of the ''Biilish Thennal UmL" is 777 8 foot-pounds, and that of the Kilo-C'aloiv is 120 7 kilojiiaiu- meti es ' . The mechanical equivalent of heat enlcis mlo many ol Ihe loi- mulas of thermodynamics It is often called Joule's Equivalent, and is geneially represented by the symbol ,/ The symbol .1 is used foi the iccipiocal of Joule's eqimalenl, or ]/,/ 15 Scales of Temperature In the constincliou of an oidin- aiy theimometer a fine tube of unifoim boic is chosen, and a bull) is formed on it to contain the mercury oi othci liquid whose expan- sion is to be u^ed as an indication of tempciatmc When it is filled the two fixed points aiedetcimmcd by placing Hie msliiimcnl (a] in melting ice, and (b) in the btcam coming fiom water boiling iindci apiessureof one atmospheic. The position lakcn by I he end ol the column of liquid in the tube is maikcdfoi each of (hese two points The distance between them is then divided into equal pai Is winch aie called degiees.lOOpaits foi the Centigrade scale and ISO for the Fahrenheit scale. By this constutcLion equal steps in tempeialuie aie denned by equal amounts of expansion on the pail of the * In absolute (c g ) units the gramme caloiy will be taken in this book IXH equivalent to 4 1SGS x 10 7 Digs, 01 cm dynes I FIRST PRINCIPLES II :lccted liquid, or ralhei by equal amounts of difference between lie expansion of Lhc liquid itself and that of the glass in which it . contained, for it is the diffcicncc of expansion that detcimines ic rise of the column in the tube This common method of leasurmg temperatmc gives results that vaiy for diffeient liquids nd foi different soits of glass Each of two mercuiy thermometers, n example, ma}' have the fixed points conectly marked, and be f unifoim boic, and -\ct if they are made of different soits of lass they may give leadings that diffei by as much as half a cgicc (Centigrade) at the middle of the lange between the fixed omls, and may show vciy scnons disciepancies sometimes mounting to as much as five degices or moic when they aie pphcd to measure highei tcmpciatures such as that of steam on its ay to an engine This illiibtiates the fact that the measmement of :mpeiatmc b\ an orchnaiy theunometei gives an arbitiaiy scale, 'Inch cannot c\ en be iclied on to be the same in difleient instru- icnts Measurements of tcmpciatiue aic much less capncious if we ~'lcct loi the expanding substance an^ one of the so-called peiman- nl gases, such as air, 01 mtiogcn, 01 h\diogcn, taking caie oi ouisc to keep I he picssine of the gas constant \\lulc it is employed > incasinc tcmpcialuic by its changes of volume Such an mstiu- icnl is called a constaut-picssuic gas thcimometei It v^oulcl be icoiu ciucnt loi oulinaiy use, but it sci\cs to supph a scale \\ith 'Inch Ihc leadings of an oulinaiy Ihciniomclci can be compaied 'luis Lhc leadings of an\ inciciny thcimometei can be collected to iiiii> Hum inlo agiccnicnt with the scale ol a gas theimometei if ha I scale be adopted a-, the standaul scale in stating tempeiatmes E\pciimcnls on the expansion of \anous gases by heat ha\ e hown lhal all gases which aie fai fiom the conditions that would ausc liquefaction expand vciy neatly alike Thus if we compaie an ir Lhci inonieler with a nilvogcn ot a hydiogen theimometei MC get iiaclically the same scale except at extremely low tempeiatures uch as those at which the gas is appi caching the liquid state, iascs expand by almost exactly the same amount between the wo fixed points, namely by 100/273 of the volume they have at the cmperatuic of melting ice , and at mtcimechate points, 01 at points icyond Lhc range, then agt cement with one another is almost )crfcct Hence the scale of the gas thermometer is much to be (referred to that of any mercury theimometer as a means of stat- ng temperature. But there is another and even stronger reason for 12 THERMODYNAMICS [cn. this pieference We shall sec later thai it is possible 1o imagine a scale of tempciatuie, based on geneial theimodyiianuc pimcipks. ^ which does not depend on the properties of any paiticular sub- stance that scale is called tne thci modynamic scale of tempeiatmc, and much use is made of it in thei modynamic icasomng. The scale of a gas theimometer is piactically identical with the theimo- dynamic scale Taking the hydiogen theimometei, in which the agieement is closest, Callendar has shown ' that midway between the fixed points the scale collection (that is, the difference between the numbeis \vlucli state the same tempciatuie on the Ivydiogcn scale and the thermodynamic scale) is only 000] 3 of a clegicc, and that the tempeiatuie has to go up to about 1000 01 down below - 150 belbie the collection becomes as much as 1 of a ^ - degiee These figures are foi hydiogen expanding undei a constant J piessuieof oneatmospheie The differences between the scale of the gas theimometei and the theimodynamic scale aicc\ en less if a con- stants olume type of gas thermometei be used, in which increments of tempeiatuie aiemeasmed by the inciementsof piessmcthat are lequued to keep the volume of the gas constant vihile it is heated 16 Reckoning of Temperature from the "Absolute Zero " Experiment shows that the amount by which an 01 ludiogcn or any othei so-called "permanent" gas expands between Lhe two fixed points that is to say in passing fioni the tempera lure of melting lce to that of boiling watci (at a piessmc of one atuio- j spheie) is about 100/273 of the volume at the lo\\ci fixed point, l caie being taken that the piessuie does not change Hence if ^e adopt the scale of the gas thermometei as our scale of tempcia- rwr tine, and use Centigiade dnisions, this icsult may be e\picssod ' ' by saying that when 273 cubic inches of gas at C aic healed imdei constant piessuie to 1 the \olume altcis to 27 I cubic inches When the gas is heated to 2C its volume becomes 275 cubic inches, and so on. Smulaily if the gas be cooled horn C to - ] C its \ olume changes fiom the onqmal273cubic inches to 272, and so on Putting this in a tabulai ioim, kt the volume be 273 at C. It will become 272 at 1 C and finally would be at 273 C H P^ Ca I) enda ^' nt!ieAermod ^ aiulCftlc oc Phys boc vol svm, oc Plul Mag Januaiy, 1903 FIRST PRINCIPLES 13 1 the same law could be held to apply down to the lowest tempeia- iiies. Any actual gas would change its physical state befoie so low temperature weie reached, becoming fiist liquid and then solid, nd the volume to which it would contract would consequently be ot zcio but the volume of the substance in the solid state. The above icsult may be concisely expiessed by saying that if jmpeiature be reckoned not fiom the oidmaiy zeio but from a 210 which is about 273 Centigrade degrees below it (moie exactly 73 1), the volume of a gas, heated under constant piessure, is roportional to the tcmpeiatiue icckoncd from that zeio. The cio in question is spoken of as the Absolute Zeio ot tempeiatuie )enotmg any tcmpeiatiue on the oidmary scale by i and the oiiesponding lempeiatuie" reckoned from the absolute zeio by T, r c have (using Ccntigiade degrees) T = t + 273 1. The absolute zcio has been defined hcie by lefeience to the ex- >ansion of a gas Hut it will be been latci that the theimoch namic c-alc of lempciatuie blaits fiom a xeio which is absolute in the disc that no lowei tcmpciature can possibh exist, and that the eio oi I he Ihcimocl^ namic scale coincides \\ith the zeio of the gas calo as defined abo\c' 17 Propeities of Gases Charles' Law and Boyle's Law. 'he experimental fact that all pumanent" gases expand b\ veiy it'aily I he same ii action of then \olume toi a gn en mciease of empeiaLme, the pussuu- being kept constant, is known as Chailcs' ,aw Anothei fimdamcnlal piopeih of gases, disco\eiecl by the xpciimcnts oi Boyle-, is thai when the volume ot a gas is alteied by Iteimg the picssme, the tcmpeiatiue being kept constant, the 'olume vanes mvciscly as the picssme Thus if V be the volume of a given quantity of any gas, and P he piessmo, then so long as the temporal me remains unchanged, r vanes mvciscly as P, 01 PV = constant This is Boyle's Law. I is, vciy neaily though not exactly true in gases such as an or ixygen 01 mtiogen or hydrogen the deviations fiom it are very li<>ht m anvgas that is m conditions fai icmoved fiom those which )i educe liquefaction * Tho e\aot position of tho absolute zero is uncoitam to the extent of about no tenth of a degree Callendar places it at -273 1 C That figure is used in his oteimmations of the pioperties of steam, and is adopted in this book 14 THERMODYNAMICS L c " 18. Notion of a ''Perfect" Gas. In dealing with the pio- peities of gases and with the theimodynamics of heat-engines it is com enient to imagine a gas which exactly confoims to laws that aie only vciy neaily tiue of ical gases. Such a gas is called a "peifect" gas The piopcities ol'ieal gases aie mosL easily treated as small deviations i'lom those oi'imaginaiv "peifect" gases obe} r - ing simple lav\s Among ical gases hydiogen piobably comes neaiest to the ideal of a peifect gas., but no leal gas is in this seme stuctly peifect. In a gas which is peiiect in the sense of coiifoimmq exactly Lo Boyle's Law we should find PV strictly constant, so long as the Lcinpeiatmc is constant If we define the Umpei.ituic scale by refciencc to the expansion of the gas we should also have V vaiymg as the tempeiatuie T (icckoned fiom the absolute >icio) undei any constant pressinc Combining these two statements we should have PV = RT (1), wheie R is a constant We may wiiLe, ior any gas assumed to be pciJ'ecl, wheic P n and V Q aie the piessiue and volume aL C When Ihc volume is leekoned pei uml quantil v of Lhe gas \\L luu c a dehmte constant value oi R f 01 each gas, depending on Ihc units employed and on the specific, density of the gas in question It should be noticed that Avhen a gas saLislyni" this equation is heated nuclei constant piessme and conscqucnth expands, R i-, a measuie of the amount of woik done by the gas in this expansion for each dcgiee tlnough which the tcmpcialmc uses Let the- oiigmal teiu[)eiatme oi Lhe oas be r l\ and its volume V-^ and let il be heated undev constant piessuic P till the tcmpcialuu is '1\ and the \ olunie 1 T , Then we lun e R r l\ = Pl\ and Rl\_ = PV ^ horn which R ( r J\ - r l\] = P(V, - rj, which is Lhe woik clone In Ihc gas m expanding fiom V\ to F 2 Let the mlei val oftcmpeitituic be 1, then R is equal to the woik clone. Thus 7? is mimevicalh expiessecl in units of woik pei unit of mass and per clegicc in loot-pounds pei Ib 01 in kilogiammcties [)Ci kilogiammc If w r c use the Centigiade degiee in both cases the latio of the numbei which expiesses J? in foot-pounds pei Ib to the numbei which expresses it in kilogiammetics per kilogramme is 3-280S5, namely the numbei of feet in a metie Accoi cling to measurements by Reguault a cubic metre of dry an , J FIRST PRINCIPLES 15 it a temperatme of C. and piessure of 1 atmosphere as defined n Ait. 12. contains 1 2928 kilogiammcs. We should accoidmgly lave for dry air, il it weie "peifect," It = I 03274 x 100 2 /1 2928 x 273 1 = 29 25, n kilogramme Ires per kilogiamme, at the latitude of London. The acLor 100- is requned to conveit the pressme into kilogrammes per quare metre. The coriespondmg value of R in foot-pounds per b. is 96-0. In this calculation air is Lieated as if it confoimed exactly to ioyle's Law For the present it is to be midei stood that the ymbol jT stands for tempeiaLuienieasuied on thescaleof a gasthei- iiomctei, liom a zeio which is 273 1 below the melting point of ice 19 Internal Energy of a Gas: Joule's Law The Internal tnogy oj a given quantity of a gas depends only on the tempeiatme. This is an inleience fiom the fact established b} T experiments of ouk that when a gas expand* without doing external too? A and ^ithoul lahing in o> giving out heat, and tliaefoie without changing ',s -i/06'A of inteinal eneigy, its tempo ahue does not change Joule's Law is to be regarded as stnctly tine onh of nnaginaiy ciitct gases m any actual gas theic i^ a slight depaiUne liom it, . Inch is \ ci3 r small indeed in a nearly peiicct gas such a^> hychogen 'he law u r as onginally established b> means oi the following xpeinnent. Joule connected a \ essel containing compressed gas uith anothei csscl winch was emph , In means ol a pipe \\ith a closed stop-cock loth vessels \veic mimeiscd in a balh ol watei and ^eic allowed to ssinne a unifoim temperature Then the stop-cock wat, opened, nd the gas distiibutcd itself betueen the t\\o \ r csscls, expanding ithout doing exteinal \voik Aftei Urn the tempeiatuie ol the r atci in the bath was found to have undcigone no appieciable liange The tempeiatuie of the gas appealed unalteied, and no eat had been taken m or given out by it, and no woik had been one by it. Since the gas had ncithei gamed nor lost heat, and had done no oik, its internal eneigy Avas the same at the end as at the begin- ing of the expciiment The piessure and volume had changed, but ic temperature had not. The conclusion follows that the internal icrgy of a given quantity of gas depends only on its temperature, ad not upon its pressme or volume, in other words, a change of ressure and volume not associated with a change of temperature 16 THERMODYNAMICS r T does not altei the internal energy Hence in any change of tempera- ture the change of internal energy is independent of the icl.ilion of piessure to volume duimg the operation it depends only on the amount by which the tempeiatuie has, been changed The apparatus used by Joule in thib cxpcinnenl is shown in fig. 3 The vessel A was filled with an compicssed to niou Ili.in 20 atmospheres, and B was exhausted. Both \ csstls were nnmeised in a bath of watei The watei in the bath wns slmul and I lie tempeiatuie noted befoie the stop-cock C was opened Allc'i Hie oas had come to lest in the two vessels the water \\:is again si med, O and was found to ha\ e the same tempuatmc as bdoie, so l.u .is tests made by a veiy sensitive thei- mometei could detect. In another foini of the appaiatus Joule sepaiated the bath into three poi turns, one poition lound each of the vessels and one lound the con- necting pipe When the stop-cock wa? opened the watei suiioimding A vas cooled, but this was compensated bv a use of tempeiatuie in the uatei suiioundmg B and C. The gas in A became coldei 1 in llu ael <>l expanding, but heat was on en up in B and C'as Us eddying niolion settled down, and when all uas still thcie was nulliei nam noi loss of heat on the whole, so fai as could be delected in I Ins loim ol expenment. It is now, ho\\c\ ci, known that a very slight chani'f ol lunpc ia- tuie does in fact take place uhcn a ical gas expands wilhoul domn uoik In latei expeiimenb, by Joule and Thomson (Loid Kelvin) a moie delicate method was adopted of detcclmi> ulic.llui llu le is any change of intemaleneigy when thepiessme.ind volume cliani>< undei conditions such that external woik is nol done Tlie L>as was f oiced to pass tlnough a poioiis plug by nijiiiiliinini> <i conslanl liigh piessme on one side of the ping and a consUinl low pussniv on the other Caie was taken to picvcnl ,'inv hc.it lu-mg gnjncd 01 lost bv conduction fiom outside In this opeiation AVOI k w.is done upon the gas m foicmg it up to the plug, and M'oik was done by i( when it passed the plug, by displacing gas undei the lower pic ssine on the side beyond the plug If no change of U-mpciahne look place, and if the gas confoimed to Boyle's Law, those two quant ilies of woik would be exactly equal, and consequently no e\lenial woi k ij FIRST PRINCIPLES It would be done on the whole. For IcL P be the pressure and V^ the volume before passing Lhe plug, and P 2 the piessure and V z the volume after passing the plug, the %olumes being in both cases slated per Jb of the gas Then the work done upon the gas (pei Ib.) as it appioaches the plug is PJ'i, and the woik done by it as it leaves the plug is P^V^ If the Lempeiature is the same on both sides these quantities are equal in a gas foi which PV is con- stant at any one tcmpeiaUue Thus a gas which is ' peifect" in the sense that it confoims stuctly both to Bo} le's Law and to Joule's would in its passage of the plug have expanded without (on the whole) doing any woik, and theiefoie without changing its internal cneigy, no heat being gained 01 lost In such a gas no change of tcmperatiue should accordingly be found, as it passes the plug, and if a change of tempeiatme is observed in any leal gas it is due to the fact that real gases aie not stuctly '" peifect." In the expcinncnts of Joule and Thomson' small changes of tem- peialuic wcic in lacL detected and mcasuied in an and othei leal gases, on passing the poious plug This Joule-Thomson effect, as it is called, is mgencial a. cooling Obsen ations of the Joule-Thomson cilVct aic of great value in dcleimimno exactly the piopeitics ol gases and vapouis which aic not peifeet, anel (as \\L shall sec latei) ceufain [uaehcal methods of hquei\mg gases undei extiemc colel depend upon the existence ol this c licet In the uuaginaiy peifect gas, ho\ve\ei, the Joule-Thomson ellcct is rnlnely absent Theic is no change of tempeiatuie in passing Ihc plug, and thcic is also no change ol internal cncigN , foi no woik is done and (In assumption) no heat is Liken in 01 givenout II is unpoitant lo notiec that \\e assume the. imatniiaiy peifect gas to satisfy two conditions it obc^s l$o\ Ic's Law exactly and also Joule\ Law exactly These chaiactciistics aie independent of one another it would be ])ossiblc to ha\c a gas satisiy one anel not the olhcj, but a gas is said to be peifect in the thcimod} namic sense onlv ^hcn it satisfies both, and in that case ceitam othei piopertics follow which will now be pointed out 20 Specific Heats of a Gas. The Specific Heat of any sub- stance means the amount of heat requncd pel dcgicc to laise the tcnipcratuie of unit quantity of the substance, undei any assumed mode of heating. Thus when a substance is heated tlnough a small interval of tempcratuie dT the heat taken in (per Ib ) is * See Kelvin's Mathematical and Physical Papers, vol i, p 333 H T 2 18 THERMODYNAMICS i<-u KdT, wheie K is the specific heat for Lhe parLiculai eondilions and mode of heating. In dealing with gases or olhcr Hinds two impoitant modes of heating must be distinguished we may heal them undei conditions of constant piessuie 01 oi consLjuii volume We shall use the symbol K p to icpiesent specific beat at conslanl pressme, and K to repicsent specific heat at constant \ olume Considei first the opeiation of heating unit quantity of n peifecl gas at constant volume, fiom tempeiatuie r J\ up to lcm[)eialuie T 2 . The heat taken m it> Jr /T <. **-(* 2 ~ ^ !> No external woik is done, for the volume (by assuni])lioii) docs not change, and consequently all this heat goes lo muease (lie stock of internal eneigy contained in the gas. But by Joule's Law the internal eneigy depends on!) on the tcmpcratuic Thueloie if we heat the same quantity of the same gas in any olhei m;mnu fiom TI to J" 2 , the same change of mteinal cncigy must take place Suppose then another mannei of heating, namely at constant piessuie In that case the heat taken in is K P (T*~1\} Dunng this piocess external woik is done, because the gas ex- pands, and its amount is P ( V V \ * \ f 2 ' 1JJ wheie F, and F, icpiescnt the volumes al Ihe beginning {U icl end of the opeiation iespecti\ ely, and P is the piessme, which by assump- tion is constant Since PJ\ = RT 2 and PJ\ = R2\, we may \vi ilc the expiession foi the evteinal woik in Ihe foim This is in woik units in heat units it is wheie ^4 is the lecipiucal of Joule's equivalent (Ail 1 I) The dilfeience between the heat taken m and I he woik done is simply an addition to the stock ol mteinal enemy Jlnl 'is was pointed out abo\ e, the change of intcinal cncigy must be Uie same in both modes of heating, and theiefoie K v = 7v 7) AR . (>>) This impoitant relation between the two specific heals m a pcifecl gas follows fiom the Laws of Boj lc and of Joule We haxe heie taken K v and K, as apply mg thioni-hout a finite ij FIRST PRINCIPLES 19 range of temperature from T to T 2 . But this range may be made mlinitesmiaJly small without affecting the argument* and in that case K v and K v become the specific heats at a definite tempeiature. The conclusion holds that for any condition of the gas K p ~ K v = AR, and this is tine whether the specific heats aie or aie not inde- pendent of the tempeiatme. 21. Constancy of the Specific Heats m a_ Perfect Gas. Fiom the above icsult it follows that if either of the two specific heats is constant the othei must also be constant. To be constan specific heat has to be independent both of the press me ant the temperatme. Fust as to independence of pleasure- we have seen (Art 19) that the internal energy of a peifect gas depends only on the tempeiature and is independent of the piessiue. If we heat a peifect gas tin on oh 1 at any one tempeiatme the change of internal oneigy is measmed (Ail 20) by K v , no mattei what is the piessme. Hence K v is independent of the piessiue, and since, by equation (2), K v is equal to K v + AR, it follows that K p also must be independent of the pressure. But a gas may confoim to the La\\b of Boyle and Joule without laving K p and K v independent of the tempeiatme, and if \\e. aie Lo lical them as constant we must make a duthci assumption "cgaiding the piopeilies of that com cnicnl imagmaiv substance a xi feet gas Rcgnault's cxpciiments sho\\ed that in some gases /v }) is icaily constant tlnoui>h a modciate range of tempeiatme But it is low known thai in most gases the specific heat becomes distinctly >ieatci at high tcmpcratuics This vanation will be discussed m I'hapk'i VI, for oui picscnl pin pose it Avill simplify matteis to think >1 au ideal gas m which the specific heat is constant. Accoidmgly, 11 dealing with a peifect gas, it is assumed that K p in such a gas is ti icily independent of the tempeiatme This is a thud assumed " Suppose the heating to bo iluough a vciy small mfceival of teiupoiaLiue dT n licatm^ at conslanL vohuno, the heat taken in is K dT, and all oi it goM to iciOdBo the intoinal oncigy bv an amount dE Hence K v dT=dE a hoatini; tiL constant piessino througli the saiae interval of temperature the eat taken m (dQ) does work dW and also adds to the internal energy by the mount dE dQ is K t ,dT, and dW is PdV, which is equal to MT Ilence iom winch K IS -K V =AR 22 20 THERMODYNAMICS [en. charactenstic ol a peitcct gas, additional to the two already dc- scubecl in Aits 18 and 19 It docs not in any way conflict uith them each of the thicc chaiacteiistics is independent of the otheis. With this finthcr assumption we have, for any pcifecL gas, K p constant under all conditions, and consequently K v also constant undci all conditions, since the diffeicnce between them is constant. 22. Reversible Actions. We ha,\c now to consider pailiculai modes in which a woiking substance may be expanded 01 com- pressed and may take in 01 give out heat, and at the outset it is important to distinguish between actions that aic teveisiblc and those that aie n reversible An expansion or compression is icveisiblc if it is earned out in such a manner that the opeiation can be ic versed, with the lesull 1 that the substance will pass back tluough all the stages thiongh winch it has passed dining the expansion 01 compicssion and be in the same condition in all respects at each coiiespondnig stage in both processes. This implies that the substance must expand smoothly, without setting up a,ivy motions within itsclL of a kind such that then kinetic eneigv is fntteied clown into heat through intcinal fuction The whnls and eddies which occui as a fluid entcis or expands in I he cylinder of an engine arc mcversiblc, and in ideal icvcrsiblc cxpan sion we must suppose them absent. Reversible expansion implies that there arc no losses of mechanical eflecL from any soit ol inlci- nal fiiction It excludes thiottlmg, such as occuis when a sub- ' stance expands Lhiough a valve or other constricted opening into a legion oi lowci press me whcic the kinetic cneigy of the sticam ' and eddies is dissipated. In such cases the motion of the sticam and eddies cannot be levcrsed. To get the subs lance back to the legion of highci piessurc would icqunc an expcnditme of mote woikthan was done upon it during its expansion, and if we wi-ie to loi.cc it back we should find it had gained heat tlnoiigh I he subsidence of the internal eddying motions, though no hc r it had come in iiom outside. The kind of expansion winch takes place in Joule's expeiimcnt (Ait 19) is an extreme instance o[ nievcisiblc expansion. ' A transfei of heat to oi from any substance is rcveisible only il the substance is at the same tempera tin c as the body fiom which it is taking heat 01 to which it is giving heat. Suppose, for instance, that a substance is taking in heat iiom a hot souicc and is expanding as ij FIRST PRINCIPLES 21 it does so The expansion may be leversible in itself, that is to say it may involve no internal fnction, but unless the temperature of the substance be the same as that of the souice, the operation as a Avhole consideied in its i elation to the souice cannot be icveised. So consideied it is reversible only when the fuither condition is fulfilled that compiession of the substance willieverse the biansfer of heab, giving back to the source the heat that was taken from it. An}'' Iheimal contact between bodies at diffeient temperatures involves an mcversible tiausfer of heat. Neither the expansions and compiessions noi the tiansfers of heat that occiu in a ical engine aie ever stnctly reversible, some of them indeed aie fai fiom being levcisible. But the study of an ideal engine, in which all the operations aie icveisible, is of fundamental importance in the science of theimodynamics, and it fiunishes a basis foi the cntical analysis of actions in a leal engine. 23 Adiabatic Expansion. Thcie aie two specially impoitant kinds of icvtisiblc c\[)ausion, (1) Adiabatic and (2) Isothcimal Achcibalic expansion 01 compiession means expansion or com- pression, earned out icveisibly, in winch no heat is allowed to cntci 01 leave the substance A cmve drawn to show the i elation of picssurc to \olume dining the piocrss is Ccilled an adiabatic line Adiabatic action would bcieahzcd if \\c had a substance expanding, 01 being compiosscd, without change of chemical state, and without any eddying motions, m a CN lindci \\hich (along with the piston) was totallv imperious to heat Fiom Ihis definition it follows that the noik which a substance docb while it is expanding adiabaticallv is all done at the expense of its stock of inleinnl cneigy, and the \\oik which is spent upon a substance when it is being compiesscd achabaiicalK all goes to i no case its stock of internal cneigy. In actual heat-engines the action is never stnctlv adiabatic, foi Lhcie aie alwaj s some exchanges of heat between the -\\oikmg sub- stance and the sin face of the cylmdei and piston Very lapid com- piession 01 expansion may come near to being adiabatic by giving little time for any tiansfer of heat to occur. After what has been said aheady about icveisibiht}', it is scarcely necessary to add that expansion thiough a throttle-valve is not adiabatic, though it may be (and generally is) done without letting heat enter or leave the substance In the adiabatic expansion of any substance work is done, and 22 THERMODYNAMICS [en. since no heat is taken in or given out, there must be a deciease of internal energy equivalent to the amounl of the woik done by the substance Taking the general equation (Art 9) tlQ = AdW + dE, winch applies to any small change of stale on the pait of any sub- stance, we have dQ, = when the action is adiabatic, and hence foi an adiabatic expansion AdW = - (IE. Ilcie dW is the woik done, A is the factor icqiiircd to convei t an expression foi woik mLo heal units (Ait 1 1), and dE is the change of internal eneigy 24 Isothermal Expansion. Isolhcimnl expansion or com- piession means expansion 01 compression canied oul icvcisibly (as regaids inLcin.il aclion) and witliouL change of lempcialuie A cuivc diawn to show the icLition of picssine lo volume dining isothcimal expansion or compicssion is called .in isothennal line When a subslancc is expanding isolheun.illy iL lakes in heal to maintain ils tcmpeiatmc consLant , it Uu-icfoic mnsl be m con lac I wilh a souicc of heal. When il is beinn compicsscd iso- theimally il c;i\cs oub heat, and inusl ho in conlacL ^v r llh ,1 icceiv ci which can lake heal fiom it 25 Adiabatic Expansion of a Perfect Gas. Considei nexl the bchavioui of a peifccl i>as dining adiabalic expansion 01 coin- piession. We have seen that in a small aduibalic expansion ol any subslnnce (Art 23) In a peifecL gas dE = K v dT (AiL 20) Hence in the adiabalic expansion of a peifcct gas But P = RT/r (Ail IS) Hence ARTdP/I r +K dT=.0, 01, dividing by T, ARcW\V 4- K v clT/T = 0, which gives on mlcgrotion AR log e V + K v log c T = constant (3) FIRST PRINCIPLES 23 Writing K v ~ K v for AR (Art. 20), and dividing by K v , which is constant (Art. 21), (K,/K V - 1) log e V + log B T = constant. We shall wntc y for the ratio of the two specific heats, namely Thus we have y log, V - log c F + log e T = constant Fmlhci, since PV/T is constant, log, P + log e y log t T = constant Adding Ihese two equations log, P -f y log, F = constant which ives PF y = constant .. ( 1). (5), (6) as I he equal ion of any adiabatic line in the piessiue-volume diagiam, to i the adiabatic expansion of a peifect gas- 1 26 Change of Temperature m the Adiabatic Expansion of a Perfect Gas When a gas is expanding adiabatically its block cjf inUinal eiK'iy\ is, as we ha\c' seen, being icdiicc-d, and hence its Unipciatuie falls, the change- (jf inteinal eneigv being piopoi- Imiul to the change of Icnipeiatine (Ait 20) Coiuciscly, in adi ibahc conipiession the tempeiatnie uses The amount bv \\lnch the Icmpciatmc is changed (in a peilect gas) ma> be tound b\ r eonil)ining the equations i\r l v = p i vj and PJ^IP.V, - T.IT, Multiplying them togelhu vc ha\e , I , , r whence =- ( =- ) , 01 1 2 = 1 1 J l V 1 This icsult of couise applies to compiession as well as to expansion dong an adiabatic hue It may be got dncctly fiom equation (4), ,vhich can be wnttcn log.T-l- (y - l)log e F= constant; whence Tj^- 1 - constant . (7) Combining equations (0) and (7) and eliminating F, we obtain y-\ he lurthci relation T/P * = constant * Itiatoborcmombeic(lthatlog e ,the' hyperbolic" 01 ' Napicnan" or"natuial" iganthm of any number, is 2 3026S5 times tho common logaiilhm of the number. 24 THERMODYNAMICS |cn. 27 Work done in the Adiabatic Expansion of a Perfect Gas. In any kind of expansion ot any fluid the ivoik done in expanding fiom volume J\ to \ohunc V, is ir = If the natuie of the expansion be sueh that PV n is constant, n being any index, then P at any point when the volume is /" is P 1 F 1 "/F", P 1 and F t being Ihc picssinc and \'ohinH in UK initial state. In that case, ioi expansion Jrom Vj lo /' ',, W = PJV I dF/F", vlncli gn es on intcgiation W = p Y " (V , 1 ~ n V 1 ~")/(1 n) , P V P V or jp^-A-J. ^ 2 (8) So fai \\ehavemadcnoassumptionas lo llic inline ol Hie \\oikmu substance Apph this icsiilt to a gas expanding adialulieallv, Ioi \\lucli the index n is equal to y (In Eq 0, Ail 2.;) \\ r c I hen Ii.ivi- W = F ^^*J^ r ^ T ^ ( c,) y 1 y 1 since P 1 J r ,= li r l\ md PJ',- A"/', Fuithci, it follows horn Ait LM Hut llns expiession (mul- tiplied b\ A) is the decicasc of mlunal cncigy piodiucd b\ llu expansion. 28 Isothermal Expansion of a Perfect Gas In ,i o,,s wind, satisfies the equation PV = JfT, Pl r is eonslanl (lining isollu iin.il expansion 01 completion, and an\ isolheini,il line on I he JJK-SMIK - volume chagiam is a lectangulai hvpcibol.i. Hie pu-ssiuc \aivin- imeiseh as the \ohinie To find the uoik done m the isolhum.il expansion of a ..as tiom Fj to J\\\e ha^c , IV = PdV r, and p = pjrjp fiom which ]y = ^ ir FIRST PRINCIPLES 25 tegiatmg, W - P,V, (log. F 2 - log e F,), 1 stead of writing PjFj we may wnte PF, since the pioduct of P d F is constant tlnoughout the piocess, and again, since r = RT, V W = RT\o^=^ . (10), 1 icie T is the tcmpeiature at vhich the piocess takes place, is cxpiession sen es to give eithei the woik that is done by a 3 in isotheimal expansion 01 the woik that is spent upon it in Lhcimal compiession Dining the isotheimal expansion or compiession of a perfect 5 thcie is no change of internal eneigv, since theie is no change tenipciatuie and the internal eneigv depends only on the npeiatme (AiL 19) Hence dining isotheimal expansion a ifect gas must take in an amount of heat equivalent to the ik it does, namch JA'2 1 lot^ r,/F J9 and dining isotheimal com- -ssion iioin F, to J\ it must gn e out that amount of heat 29 Summary of results for a Perfect Gas It ma\ be con- ucnt al llus poml to collccl the icsults that ha\ e been found ac Lions occiuimg in ]ieilcct gases it is assumed that the gas satisfies Bovle's La^ (Ait 17) and ulr's Law (\il 10) and that the specihc heat (at constant ssuie) is mdopt'ndc'iil of the tcmpciatuu Fmthci, the tem- aLmc is mcasmcd on llic scale tuimshcd b> the c\])ansion ot gas ilsclf Undci I lie-so conditions Me ha\ e the following I'lLs pv - RT, CK- 7i' is a constant depending on the specific density ot the gas , K^-K^AE, cic K v is the specific heat at constant piessuie, K v the specific it al constant volume and A is the icciprocal of Joule's equn a- l. K 1t and K v aic both constant n axli aba tic expansion PW = constant, or PjPo^ (V^JV^, ei c y is K V /K V TVy-*= constant, 01 TJT Z = (FJF^v- 1 . y_l 7-1 T/P v = constant, 01 T^T,;. = (Pj/P 2 ) ? . 20 THERMOD YNAMICh Heat taken in =-= 0. w . , R^-TJ PJ\ -PJ r , Work done = v l - i - y- 1 y- 1 Deci ease of Internal Eneigy "^ J J J y \ In isothermal expansion PV constant, since T Heat taken m = , , Change of InLeinal Kncigy = 30 Fundamental Questions of Heat-Engmc Efficiency Wt are now in a portion to deal \viLh MK mosl ['imdim<nl.d questions of htat-cngine efficiency, which m.iv In si. tied ni lh< following tenns (1) HaMng gi\en a ;,oiiicc 1'ioni ulncli lu-.il in.i> IK l.ikcn in at a high lempeiatiuc, and a sink 01 uci-ivi r to \\liu-li lu.il m.i\ be lejected at a lo\\cu tcnipcialiuo, ho\\ 111.1^ IK .il [.dun IK. in UK souice be utilized to the bcsLj(l\aiil,n.cloi llu |)iii|xis( (.1 |Hodiicin-. mechanical effect ? In olhci \\oids, Jio\\ ni.i\ (he yu.iicsl .iiii<.iin[' of \\oik be done b> each unit ot lit-.il l.dvi-n liom llu hu[ S.HIK, f (2) What fraction of the he.il Lakcn (mm (| u hoi somcc is il theoietically possible tocomcil into \\oiki 1 Jn olhci wt.uls, \\li.il is the limiting efficiency of convcisicm ? 31. The Second Law of Thermodynamics So l.u .is llu- Fnsl LawofTheimodynamics(AiL S) o OCS , il isnot ol>\ ions I h.,1 (luicis anything to pievent all ihchenl u Inch I he somcc c.ui snppK I, on. being- convex Led into woik. Cut iL will j)u-si-i.[]y l sn n (h.il a limit is imposed as a consequence of Lhc following pmiciple uhu-h is known as the Second Law of Tltnmodijnan,,^ It u impossible foi a &clf-actn,g machinr, nnm<lr,l In, ,, nltnial agency, to convey heatf, om one body to unotha at a /,/;/-, /, ,, , tlltu , The beconcl Law says, in clfccL, that l.t-.iL will n,,l p,,ss automatically from a colder to a hollc-i body. \Ve can I,HC-- .[ to pass up, as in the action of a rcliiyc-iutmft madunc , !>( ,,| v l,x applying an ^exteinal agency" to dnvc the , nm | llll( .. A hr,,l- engine acts by lethng heat pass down from a ho, u i 1,, a oolcU- body, not of couise by duect conduction ftoin one Lo lh, other, l m - llu FIRST PRINCIPLES 27 i mode of tiansfer in which the heat would do no woik, but by king the working substance alternately take in heat fiom the , body and icjecl heat to the cold body, and thcieby undergo mansions and contractions in which its pressuie is on the whole ater during expansion than dining contraction, with the lesult it a pai t of the heat that is passing down through the engine is i verted into work In consequence of the Second Law it is only eitain fraction of the whole heat supplied by the hot body that i be convcited mlo work by any such process J2. Reversible Heat-Engme. Carnot's Cycle of Opera- ns. To Ihc iirst ol the above two questions (Art 30) a coriect jwei was given by Sadi Cainot in a icmaikable essa,y s published 1821, entitled Reflexions siu hi puissance motnce du feu et sut machines piopies a devcloppu edit puissance In this essay mot maybe said to ha\ e laid the loundation of theimodynamics pointed out thai the gictitest possible amount of woik was be' obtained by letlmg the hcdt pass fioin the souicc to the ci\ei tlnough an engine wuihing in a stnctly it'vo^iblc itianne) [ onU .is ugaids ils o\\n inlcinal actions but also as icgaids ' hanslci ol heat to it fiom the souicc and horn it to the CIVLI Tlie engine conccncd by Caniot is an engine e\ci> one whose optialions is icvcisible in the sense explained in Ait 22. liuLlu'i showed how an engine might (theoietically) woik in h a uay as to satisfy Ibis condition, its cyele consisting of ;sc loin icvcisible opeiations I) Isothciinal expansion at the tempeiatuie ol the hot souice L ) Duung this opeiation heat is taken in ic\ eisibly fiom the t soince ("2) Adiabaiic expansion, dining which the tempeiatuie of the aking substance falls fiom I\ to 2 T , (the tempeiatuie of the eivei) (J) Isothcimal compiession at the tempeiatuie of the icceivei. irmg this opeiation heat is icjccted icveisibly to the icceivei. (1.) Adiabatic compiession, by which the tempeiatuie oi the .iking substance is raised fiom T t to 2\ This completes the cle by bringing the substance back to the condition in A\hich it is assumed to be at the beginning of the fhst opeiation. In the cycle as a whole work is done by the substance- the erage piessure in (1) and (2) being gieater than in (3) and (4). 28 THERMODYNAMICS [en. This cycle of operations, which is known as Cmnot's Cycle, is entnely reveisible The woikmg substance might be forced lo go thiough it in the reveised duection, taking in heat fioin the cold body and gnmg out heat to the hot body The Uansfeis of heat would be exactly ie\eised, and aL cveiy stage the piessuie and volume and tempeiatmc of the substance would be Uu- s.inu 1 as when woikmg dnect. The A\ r oik spent upon it would be equal to the woik got fiom it in the dnect acLion Cainol's idc.il en<>ine accoidingly affoids a btncllv icvcisiblc means of letting hen I down fiom the hot souice to the cold icccivci The aigument by which Carnot pioved lhat no heal-cngino can utilize heat moie complete!}' than a icveisible heal -engine ud!i/es it, in letting heat down fiom a given source lo a gn eu leceivei, is substantially as follows. 33 Carnot's Principle. To prove Lhat no oLhci heat-engine, woikmg between the same souice and icccncr of heal, can do I IK- same amount of mechanical woik as a icvcisiblc engine by Liking in a smaller quantity of heat. Suppose theie aie two heat-engines 7? and <S', one o( which (A') is icveisible, woikmg between the same hot body 01 somei ol' lic.il and cold boclv 01 leceivei of heat, and each piodncmg (h r saim amount of mechanical Moik Let Q be Ihe qiunhh of | U<1 I \\Iuch R takes in fiom the hot body. Now if/? be i excised il will b v UK e\pendituie on it of the same amount of woik one lo UK h.[ body the amount of heat it foimeily took fiom il, namely Q Foi || U s pm pose set the engine S to diivc K incised The w.iL winch N pioduces is ]iist sufficient lo dine If, and Ihc I wo niaMiincs (,V dining R) foim togclhei a self-acling nuichmc unaidtd by any exteinal agencv. One of the two, nameh ,S f , lakes lual 1'iom UK- hot body and the othci, 7?, which u, levcisible, H i Vt s back lo UK- hot body the amount ol heat Q. Now if ,9 could do Us \voik by taking less heat than Q fiom Ihe hoi boclv ll,<> hot lm\ U W //W on the -Me gam heat No woik is being done on the sysU-m fiom (- side, nor is anj heat supplied fiom olhei somccs, so whalt x n heal the hot body gams must come fiom the cold body Theiefoie il ,V could do as much woik as the icvci sible engine If, wilh a sm illc i supply of heat, we should be able to anangc a pmcly sclf-ac-lmn machine thiough which heat would continuously pass up fmn. a cold body to a hot body. This would be a A u.lalion of the Second Law of Theimodynamics. FIRST PRINCIPLES The conclusion is that 8 cannot do the same amour ith a smaller supply of heat than a reversible engine , or, r nguagc, that no other engrne can be more efficient than a re igine, when they both woik between the same two temperatuJ source and receiver Further, let both engines be leveisible. Then the same argu- ent shows thai each cannot be moie cfficrent than the other, encc all rcvcrsrblc engines taking in and rejectrng heat at the me two temperatures are equally efficient 34 Reversibility the Criterion of Perfection in a Heat- igme These results implj that, in the thermodynamrc sense, versibilrLy is the criterion of what may be called perfection in lieat-enginc A reversible heat-engine is perfect in the sense that cannot be improved on as regards efficiency: no other engine king in and rejecting heat at the same two temperatures can iLam from the heat lakcn in a gicalei piopoitron of mechanical ccL Moreover, iL this criterion be satisfied, it is, as regards efB- MICJ , a matter of complete mclifleiencc what is the nature ot the .irking subslancc 01 what, in oLhci lespecls, is the mode oi Lhe ginc's action IL is, Lheietbie, a complete answer to the fust question in Ait. 30 say that Ihe yiealcsL amount of work that is theoretically issible will be done by each unit of heaL il the heat is supplied to L engine winch works in such a wuy thaL c\ ei^ one ot its operations ie\crsiblc 35 Efficiency of a Reversible Heat-Engine The second ic'slion in Art .30 could not be answeied by C'ainot because in -, Lime lire doc-lime of the Coiisei \alioa ol Energy was unknown, d iL was not recogni/ed llia.1 paiL of the heat disappears, as heat, passing through the engine C'ainot realized that \\oik is done an engine through Lhe agency of heat, but he did not know that is done by Lire conversion ol heat It is remarkable that he was verlheless able to conccne the idea of a reversible engine and sec that rt is Lhe most effective possrble means of getting work lie through the agency of heat Ilrs argument as to tlm is per- jLly valid though rt makes no use of the JFrrst Law of Thermo- 'namrcs It rs moreover extraordrnarrly general There rs no surnptron rn rt as to the proper tres of any substance, noi as to e nature of heat, nor as to the way in wlnnh tpmnpiatm-p is to measured. All that he assumes abo JISc |inn Lib B'lore .V 1 / THERMODYNAMICS [CH. somce and the iccenei ib Lhal one is hotter than the other The aigumcnt stands by itself, and the whole passage in which it is lepioduccd heie (Ait. 33) docs not involve a leteience to any of the lesults stated in caihei Aiticleb. But foi the puiposc of ansuenng the second question of Ait. 30 we shall in the fiist place deal with one particular icvcisiblc heat-engine, namely a ic\oihible engine which has a pcii'cct gis foi ^oiking substance, and shall calculate its efficiency with I he help of the icsulis pieviously obtained foi peifect gases. Tl will be easy to go on iiom that to find a general ansAvcr to the question, What is the hmiling efficiency of any heat-engine ? 17')- Piy I- CJaiiinL'H Cycle uiih i j^as Idi woikiny 36. Carnot's Cycle with a Perfect Gas for Working Sub- stance Consulci then an ideal engine in which a subslanc-c may go thiough Ihc opciatious ol Camel's Cycle (iig 4) [uuigine a cylmdu and piston composed of perfectly non-conducling matciial, except as i eg aids Ihc bottom oi the cyhndci, whieh is a conductor. Imagine also a hot body 01 indefinitely capacious souice of heat A, kept uh\a3"> at a Icmpcvatuic r JL\, also a pcifcc'lly non-conducting covci B, and a cold body 01 indefinitely capacious icccivci of heat C, kept ah\ay^ at some tcmpciaturc 2'_j,jjv.hicli is lowei than 2\. It ijs supposed that A 01 U or C can be applied at JfJtUJNIClPLES 31 all to the bottom of the cylinder. Let the cylinder contain 1 Ib. f a peifecL gas, at tcmperatuic T 19 volume V a , and press me P a ) begin with The suffixes icfer to the points on the mchcatoi lagram, fig. 4. There aie four successive operations (1) Apply A, and allow the piston to advance slowly through uy convenient distance. The gas expands isothermaJly at T a , tak- ig in heat fiom tlie hot source A and doing woik. The pressure langes to P b and the volume to V b The line ab on the indicatoi lagiam icpicsents this opciation. (2) Remo\ eA and apply B Allow Ihe piston to go on advancing, lie gas expands adiabatically, doing work at the expense of its ternal eneigy, and the tempeiatuie falls Let this go on until the mpciatme is T z The piessuie is then P c , and the volume V c us opeuition is jepicsented by the line be. (3) Remove B and apply C Foice the piston back slowly, le gas is compiesscd usothemially at T 2 , since the smallest m- ease of tempeiatuie abo\e T 2 causes heat to pass into C. Woik spent upon the gas, and heat ib i ejected to the cold icceiver C. t this be continued until a ceitam point ^is icached, such that c fourth opciation will complete the cjcle (I) Remove C and apply B Continue the compiession, \\luch now adiabatic. The picisuie and tempeiatuie use, and, if the mt d luis been piopcily.chosjin, when the piessuie is icstoied to oiii>mal value P a , the tempeiature will also ha\e usen to its giiul A alue r JL\ [In othci \\oids, the thud opeiation cd must be j ippcd when a point d is reached such that an adiabatic line chawn oiigh d will pass tlnough a ] This completes the cycle To hud the piopci place at which to stop the thud opeiation, have (by Ait 20), for the cooling dining the adiabatic ex- usion in slagc (2), (yjV b )y-^T b /T.= T 1 /T St 1 also, foi the heating during the adiabatic compiession in - C C')' (V (l IV a )y-^T a /T^T 1 jT z . Icnce F c /F & =F./F aJ 1 thcicfoic also V c jV d = V b /V a . That is to say, the latio of isothermal compression m the third gc of the cycle is to be made equal to the ratio of isothermal )ansion in the first stage, m older that an adiabatic line tlnough 32 THERMODJNAJYLJLL2> |m. d shall complete the cycle For bie\ ity we shall dcnoU- t-illu v <>l these last latios (of isothermal expansion and compulsion) l>> / The followino aie the transfcis of heat lo and i'wm I he woilvini- Q gas, m the foui succej>si\e stages, of the cycle, quantities of heal aie heie expiessed in woik units (1) Heat taken in fiom A = m\ loy t / (I>y Ail. liS) (2) No Jieat taken in or rejected. (3) Heat rejected to C = 122' 2 log, > (by Ai I. L'S) (4) No heat taken in or rejected. Hence, the net amount of external woik done by I In- y.is, brm^ the excess of the heat taken m abo\c ilie heal re | a led in a com- plete cycle, is R (TI _ TZ ) I o ( , , this is the area enclosed by the loin cui\ es in Hie ligmi The Efficiency in this C3 r cle, namely Ihc fiac-lion Heat conveited into woik Heat taken m ' is aecoidmoly - Anothei \\ ay of bating the icsiilt is Lo sa\ lli.il il \\\ wnli Q { loi the heat taken in fiom the hot soiucc, and (^ loi Ihc lu'.il u |< d ( d to the cold leceiver, then QJTi - Q 3 /2' a . In these expie^ion-, the tempciatuics 'J\ and T, .IK iind< isl.,,,,1 to be meabiued on the scale of a pcilcct n, ls lliuinoinclM, and horn the absolute zero. 37- Reversal of this Cycle. This cycle, bcmn <t ('.H.K.I r\r!<. is le^eisible To icalize the fact moie lnll v we may consul,, m detail what will happen if we make the nnao,, 1; ,,v u>n,, l( u.,,1, backwaids, forcing it to trace out the same md.ralm <h.,u llim in the opposite oidei. Foi thi, purpose we nuisl cvp u ,d u.,,1, ,, it fiom some other somcc of woik Sta,l,,,o , IS brlmr |,,, IM [| 1( . point a (fig 4) and with the gas at I\, we shall leqnn c- 1 | u . | ( ,l!<,u ,,,,, toui operations. (1) Apply B and allow the piston to advance The o,, s , M,.,n<U adiabatically, the cuive tiaced is ad, and when d is reaelu 1 Ih, temperature has fallen to 2 1 ,. -j final .TJtUJ\Cll J LES 33 (2) Remove B and apply C Allow the piston to go on advanc- ing. The gas expands isotheimally at T 2 , taking in heat from C, and the cmve dc is traced. (3) Remove C and apply B. Compiess the gas. The piocess is achabatic. The ciuve tiaccd is cb, and when b is i cached the tempciatme has usen to Tj. (4) Remove B and apply A. Continue the compiession, which is now isotheimal at T l Heat is now rejected to A, and the cycle is completed by the cuive ba. In this piocess the engine is not on the whole doing woik, on the contiaiy, a quantity of \\oik is spent upon it equal to the area of the diagiam, 01 R (2\ T 2 ) log t ?, and this woik is conveited into heat. Heat is lakcn in fiom C in the fwst opeiation, to the amount RTo log t ? fleat is i ejected to A in the fouith opeialion, to tlie amount K2\ log c j In the fiist and thud opciations theie is no tiansfci ol heat The machine is acting as a heat-pump The action is now in c\eiy icspcct the ie\eise of \\hat it \\as before. The substance is in Lhe same condition at coiiesponclmg stages in the t\\o piocesses The same woik is nou spent upon the engine as was loimeily done b^ it The same amount of heat is now given to the hot bodj A as was fonncily taken fiom it The same amount of heat is now laken fiom the cold bodj C as was formeily niven to it This will be seen by the following scheme Cm nofs Cycle with a peijcct gas, Dnect Woik done by the gas = R (T l - T 2 ] \o^ / , Ilcat taken liom A = RT log c ? , Heat icjccted to C = RT 2 log, ? Cycle with a perfect gas. Revei&ed Work spent upon the gas = R (T^ - T 2 ) log c ? , Heat icjected to A = R2\ log c r, Heat Laken fiom C = RT Z log e 1 In the second case the heat lejected to the hot body is equal o the sum of the heat taken m from the cold body and the fork spent on the substance. This of course follows from the rmciple of the Conservation of Energy. E T 3 31 THERMODYNAMICS |CH. 38. Efficiency of Any Reversible Engine. The nnagmaiy enome, then, of Ait 36 is reveisible. Its efficiency, as we have seen, o y y * is T T Avhcie f l\ is thetemperatiue of the somce iiom which it takes heat and T 2 is the teinpeiatiue of the receiver to Avhich it i ejects heat But AVC saw, b\, Ait. 33, that all leveisible heat-engines taking m and lejectmg heat at the same tA\o tcmpciatmes aie equally efficient Hence the expiession /Tt rji - 1 ] ~ - 1 2 measuies the efficiency of any icyeisible heat-engine, and theiefoic (by Ait. 33) also expicsses the laigest fi action of the heat supplied that can possibly be conveited into Avoik by any engine Avhatcvcr opeiating between these limits In other Avoids, if AVC have a supply of heat at a tempciatme T 1 , and a means of getting ncl of heat at a tcmpcratme T 2 , then theie is no possibiht_y of coin citing moic than that fraction of the heat into woik. Tins is the mea^inc of pojecf ejjiciauij it is the theoietical limit beyond which the clllcicncy of a hcat- engme cannot go No engine can conccuably suipass this slan- daid, and as, a matter of fact any ie.il engine ialK t,hoit of it, because no ical engine is stuctlv icxeisiblc 39 Summary of the Argument I3ncfly iccapitulated the steps of the aigumcnt by which we haA r e icachcd this immensely nnpoitant icsult aie as follows Following Cainot, AVC considcied how any heat-engine woiks by taking in heat Iiom a hot somce and ic)ectmg heat to a cold iecen r ci, and established (by means of the icductio ad alisiudinn of a hypothesis Avhich Avould conflict Avith the Second LaAv of Thermodynamics) the conclusion that no engine could do this moie efficiently than a leveisible engine docs, that is to say, an engine \vlnch goes through a leveisible cycle ol opeiations Tins led to the mfeicncc that all icversible engines Avoiking between the same tcmpeiatuies of souicc and leceu ci AVCIC equally efficient, and consequently that an expression for the efficiency of any one of them Avould apply to all. and would mean the highest efficiency that is theoretically possible. Still following Carnot, we imagined a cycle which AAOulcl be icveisiblc, consisting of four stages, namely (1) isotheimal expansion dining which heat is taken m fiom the somce, (2) achabatic expansion dining Avhich FIRST PRINCIPLES 35 the tempciatine of the substance falls from the tempeiatme of he source to thai of the recen er, (3) isothermal compiession luimg A\lnch hcaL is i ejected to the receiver, (J<) adiabatic com- ncssion during \\lnch thctcmpeiatuie of the substance rises again o that of the source. Up Lo this point there had been no assump- 1011 as to ihe use ol any particular working substance. We next nquiicd what Avould happen in this c} T cle if a peifectgas weie used s woikmg substance Taking loi the scale of tenipciature a scale kised on the expansion of a perfect gas ', and expressing on this eale the tcmpciatuics of souicc and reccnci as 2\ and T 2 icspec- ivchy, \ve found that a icvcisible engine, using a peifect gas foi oiking snbslancc, has an efficiency of [cncc it was concluded that this cxpicssion measuies the efli- cney ol any ie\cisible engine woikinq between these limits, and lat this is Llic highest efficiuic}' Iheoietically obtainable in any Mt-cngmc. This gcneial conclusion inav also be staled, with equal genei- ily (foi any leversiblc engine), in ihe foini Ci/Z 1 !-^, icie C^, is the heat taken m bv the engine fioni the soiuce at r l\, d Qy is the heat i ejected by it to the veceivci at T,. The efficienc of aiu liCc\L-cninc may be wiiltcn & Q? the tnginc be ie\cisi!jle 01 not In a icveisible engine, as we may now call it, a theiniodynamically peifect engine, s becomes , m ir r 1 J. 2/J. i [n an engine winch falls short ol revcisibilny a smallei fiaction the heat supply is conveited into work and the heat rejected is itncly larger, Q.,/l\ is gieatci than Qi/^i LO Absolute Zero of Temperature. The zero from which and r 2 are measured is the zeio of the gas thermometer, which 5 defined (Art. 16) as the temperature at which the volume of the would vanish if the same law of expansion continued to apply. ' That IB to say, a scale in which the temperature is propoitional to the me ol tho gas, when the pressure is kept constant a a 36 THERMODYNAMICS [en. But we can now give it another meaning. Taking the expression for the efficiency of a reveisible heat-engine we see that if the cold icceivei weie at the lempcialuu' ol Ihe absolute zeio (so that T 2 = 0) the efficiency would be equal (<> I mothei woids, all the heat supplied to the engine uouldbecoii\( i(< d intowoik It is cleaily impossible to imagine, a icceivci eoldc i lli.ui that, for it would make the efficiency gicalci Hum 1 and llu-u li\ violate the First Law of Theimoch namics by making llu amount ol woik done gieater than the heat supplied Hence I lie /< to which we found on the gas scale is also an absolute Ihcnnodvnainie /.no, a temperatuie so low that it is inconceivable on themiodvnaim< grounds that theie can be any lowci tcmpcialine The In in "absolute zeio" has consequently acquncd a new meaning wilh- out lefeience to the piopeities of any substance we see thai il represents a limit below which tempcialuic cannot go This justifies the use of the woid "absolute" as applied lo a /eio ol tempeiatuie. 41. Conditions of Maximum Efficiency Funn llu .tho\< icsiilt it will be obvious that the avjilabilily ol lie.il loi | llin s. foimation into uoik depends essentially on tlu la.igr ol |,. mp , ,., ture thiough which the heat is let clown fiom (lul ,,| || 1( | lo( somce to that of the cold body into which heal is ie,,< i,.,|, ,| ls only in virtue of a difference of tempeiatuie be I un n bodu s I h,l conveision of any pait of then heat into work becomes p< IS sil,|, No mechanical effect could be pioduccd Irom heal, h.,w , , , ,, the amount of heat piesent, if all bodies wcio al a .1, .! \<d\ ,,| tempeiature. Again, it is impossible Lo conve.L lh, whole ol , mN supply of heat into woik, because it is impossible l, lux , , J,,,,|\ at the absolute zeio of tempeiatme as the sink mlo uhu-h ] ,l ls rejected. If T, and T t are given as the highest and lowesl I, m,,,.,,,!,,,. s of the lange through Mhich a heat-engine is to wmk , Is ,,, that the maximum of efficiency can be icached only wl,,,, |] 1( > * .s ^ o f icjecliow Temperature of icception ij FIRST PRINCIPLES 37 Any heat taken in at a temperature below T 15 or rejected at a tempeiature above T 2 , will be less capable of conversion into woik than if it had been taken in at T 1 and icjected at T a , and hence, with a given pan of limiting temperatures, it is essential to maximum efficiency that no heat be taken in by the engine except at the top oi the range, and no heat i ejected except at the bottom of the lange. Further, as we have seen in Ait. 33, when the tem- peiaturcs at which heat is icceived and i ejected aie assigned, an engine attains the maximum of eflknenc}'- if it be icveisible. It may be useful to repeat heie that in the tiansformation into kVoik of heat supplied fiom a gn en source, the condition of reveisi- 3ihty is satisfied in the whole operation fiom souice to icceiver f (1) no pait of the woiking substance is bi ought into contact lining the operation with any body at a sensibh/ chfleient tempeia- ure, and (2) theic is no dissipation of eneigy thiough internal nction The first condition excludes any unutilized drop in tem- >eiatuie, the second excludes cckhing motions and such like oinccs ol \\asle, which ausc in consequence of expansion thiough hiottlc-vahcs oi coiislncted onficcs, or in consequence of any diisc that sets up dissipativc motions within the substance n a piston and cjlmdci engine we ha\e to think ol Ihe substance s expanding by the giadual displacement ol the piston, doing /oik upon it, and nol wasting eneigy to any sensible extent by etling poitions of itself into motion Thcie aic to be no local anations of picssuic within the cUmdci, such as might occui in fast-iimmng engine lluoui>hthc meitia of Hie expanding fluid When we proceed to deal in a lain ehaptei \\ith stea.ni jets in jlalion to steam tiubmes, \\e shall sec that it is possible toha^se n Llicoiy) a icvcisible action, though the woik done by the sub- aucc in expanding is cmplojcd to give kinetic eneigy to the ibslaucc itself as a whole by foimmg a jet, because in that case le eiioioy ol Ihc jet is icco\crable when propei caie is taken to mliol Ihc foimalion of the jet But the eddying motions spoken here aic ol a diffeienl class then eneigy is niccoverable and a that icason Ihcy violate the condition of icveisibihty It may also be woilh while to icpcat heie that no real heat- igme can woik between the souice and the leceiver in a strictly vcrsiblc mannci It cannot wholly escape eddying motions it nnot wholly escape tiansfcrs of heat between the woiking sub- ance and bodies at other tempei attires. In paiticuiai, since the Drkmg substance must in piactice take in heat at a reasonable 38 THERMODYNAMICS [cu. rate fiom the hot somce, the source is usually much hotter than the substance while heat is being taken in. This is, in piactice, the most serious bieach of reversibility 111 the Lianslorraation of heat by a steam-engine. It means that between the tcmperalme of the soiuce and the highest temper aim e reached by the working substance in its cycle of operations, theie is a wasteful chop, a drop that is not utilized thermodynaimcally. If it were practicable in the steam-engine Lo avoid the drop between Ihc temperature of the furnace gases and that of the vsalcr in Hie boiler a gieatly incieased efficiency of conveision would be attainable. If we ka^e this diop out of account, and take for the upper limit 2\, not the tempeiatuie of the fuinacc gases but the temper a- tuie in the boiler, and if we also take for T the tempeiatuie in the condenser, the fraction will measinc the greatest fraction of the heat supplied to the boiler that can be conveited into woik, under idcalh favouiablc (in othci woids, strictly icvcrsible) conditions between the boilci and the condenser The perfoimancc of any ical engine falls shoit of this because it includes uicveisible features, the chief ot \\hich aic thiottling actions in the steam-passages and exchanges of heat between the steam and the mclal of the c} hnek'i anel piston. But although this limit of efficiency cannot be actually reached, it affords a valuable ciitcnon with which te> compaic the per- formance of any real engine, anel establishes an leleal J'oi engine designcis to aim nt It is important to iciih/c that a substance may expand rc- versibly although it is taking in heat from a souice hotter than rtsclf in othci woiels, thcic may be an irreversible ehop of heal between the somcc anel Ihc substance, but no incvcisiblc aclion within the snhslancc Thus Ihc fluid in a boilci is at a definite Lcmpcratmc lower than thai o( the lurnace while it is taking in heat from the fuinacc, Ihcic is accoidmgly an iiicveisiblc drop in this transfer e>l heal but the formation anel expansion of the steam may go on in a ic\eisiblc manner. We can imagine all the inlnnal actions of Ihc woiking substance to be icvcrsible, although as regaids Iransfbis of heat from the source or to the icceivci ' there is not reversibility. In thai event the engine will still woik as efficiently as possible between z/i ozou limits of lempe>atwe, JT namely the limits at which the substance takes in and i ejects |< FIRST PRINCIPLES 39 eat, though it is no longer the most efficient possible con- ivance for utilizing the full range of tempera tin e fiom source to 'ceivcr. Thus if we interpret 1\ and T 2 a*s the limits of tempeiatme of ic woikmg substance itself without any reference to a souice a receiver 2\ being the tempcratuie of the substance while it taking in heat, and T 2 the tempeiature of the substance while it rejecting heat, and if the internal actions of the substance are veiMblc, then (T l - T z )jT l still measuies the efficiency of the igmc. This fiaction still cxpiesses the gieatest efficiency that is icoictically possible in any heat-engine woikmg between the nits 2\ and T 2 When we speak of a substance as taking in heat at a stated inpciatuic, or lejccting heat at a stated tempeiatme, it is- to be tderstood that the tempeiatme of the substance itself is meant, ough that may not be the tempcratuie of the souice 01 icceivei , d wjicn \\e speak of a substance as expanding 01 being com- cssed in a icvcisiblc mannei we do not imply that it may not be Ling m heat fiom a souice hottu than itsell 01 i ejecting heat to ccc'U ei coldei Ihan itself A c\ cle of opciatioiib ma\ be mlenuilly >eisible, (hat is to sa> , ic\cisibk Y so lai as actions \\ithm tlic uking substance aic conccincd, allhough it happens to be ^cialcd with an me \eisiblc liansfci oi lieat to Hie woikmg ^stance fiom the somcc oi fiom the woikma substance to the 'CIVCl \2 Thermodynamic Scale of Temperature Rcfciencc was idem Ail 15 to the (act (lust pointed out by Loid Kch m [) that imodynaniic pimciplcs allow a scale ot tempeiatme to be uied which is independent of the piopcrlies of any paiticulai ^stance, i-cal 01 unaginaiy Up to the piesent we ha\e based the le on the piopcrties of a pcifcct gas, taking a scale m uhich dcgtecs (01 equal mtcivals of tempciatuic) concspond to ml amounts ol expansion on the pait of a pciicct gas kept constant picssuic Using this scale we have seen that a icver- Ic engine which woiks between the limits T and T 2 , and takes any quantity oi heat Qi at T 15 icjects at T 2 a quantity Q 2 equal Qi T J T i> -"d has an efficiency equal to (fj - T,)/^ " Wo 7iiay nnagino a source at T^ and icceiver at T 2 to bo substituted for the al souroo and rocoivoi, it those have a wide* lango of temperatuie, without ting the action of the Avorking substance. Mathematical and Physical Papers, vol i, p 100, also pp 233 236 40 THERMODYNAMICS [en. Now imagine that the heat Q 2 , which is icjccled b\ Ibis engine , foims the supply of a second icveisiblc engine taking in heal al T, and lejectmg heatatalowei tempeiatine T 3 , such lhal Hie inleival of tempeiatmelhiough \\hiehit woiks (T 2 - T ,) is I he same as I be inteival thiough which the fiist engine woiks (2\ - T,) Call e-aeh of these mtenals AT. Let the heat Qj icjcclcel by I Ins stcond engine pass on to foim the supply of a thud ie\ eisibh e ngme', \\oi k- mg through an equal inteival AT and icjechng heal <V, le> a loin Ib icveisible engine, and so on We imagine a scucs e>l engine s, e\ ei v one of \\hich is icversible, each passing on its ic)erle-el heal lo loi in the supply of the next engine in the scucs, and each vunknig thiough the same number of dcgiees on the peife-cl gas I he i mo- meter, AT. The efficiencies of the successive engines aie ^ AT/T 1; AT/T 2 , AT/T,, clc The amounts of heat supplied to them aie Multiply in each case the heat taken in by Lhc cllicieiiey lo Inul I IK amount of woik clone by each engine in the scnes, and \vi lincl lli.il the amount of \voik done is the same loi all the engme.s, nanu'l\ Accoidmgly, we might define the inteival e)l lempeialnie' loi each engine, without icfeience to a pcilecl gas 01 lo any ollu-i theimometric substance, as that inteival which makes evn\ engine in the senes do the same amount ol woik, and il \u- did so we should get a scale of tempeiatine which is ulenlical with (he- scale of the peifect gas thermometei. The above method of obtaining a llieimodynanuc scale of temperatme may be put thus Stalling fioni any aibihaiy con- dition of tempeiatine al which ucma^ imagine heal lobe supplied, let a series of mteivals be taken such lhal equal amounts of \\oik \\ill be done by e\eiy one of a seiiet, ol levcisible engines, eae-h woikmg with one of these mteivals foi its langc, and each handing on to the engine below it the heal which it icjccts, so lhal Ibr heat lejected by the first foims the supply of lhe> see-ond, and so on Then call these intervals of tcmpcialuie cenial VVh.it tin- above pi oof shows is that the mteivals thus ddmcd to be cepial are also equal when measuied on the scale ol I he pcilecl gas thermometer in other words, the Iheimodynamic scale and the' i] FIRST PRINCIPLES 41 pcifeci gas scale coincide at eveiy point. Any tempeiatme T icckoncd fiom zeio on the scale of a perfect gas thermomeler is also an absolute tcmpeiature on the theimodynamic scale. The conception, then, of a chain o reversible heat-engines, each woikmg through a small definite lange, famishes for the statement of temperature a scale which is really absolute in the sense of being independent of all assumptions about expansion 01 othei behaviour of any substance As the heat goes down fiom engine to engine in the chain, pait of it is conveited into woik at each step, and the lemamdei passes on to foim the heat-supply of the next engin We have only to think of the steps as being such that the amount of heat convei Led into Avoik is the same foi each step, and that the remamdei passes from engine Lo engine till all is conveited. Thus if we have n engines in the chain, and if the whole quantity of heat supplied to the fiist engine is Q } , then the steps aie such that each engine conveits the quantity Q^jn of heat into woik When n steps aie completed tlicie is no heat left all is com ei ted into work Tins means that Ihe absolute zeio of tempeiatme has been i cached we may in fact define the absolute zcio as the tempeiatme \\hich is leached in this mannci It is imagined to be icacbed bv coming down tluough a JnnLc numbci of steps of tempeiatme, each step lepiesentmg a liiulc lall in tempeiatme We define the absolute 01 theimodynamic scale by saving that these steps aie to be taken as equal to one anothei From this it will be seen that the conception of .in absolute 7oio, and of an absolute thermodynamic scale with umfoim mtcnals, docs not depend 49,11 any notion about peifect gases 01 about the piopeitics of any paiticulai substance We icaoh the absolute zeio when, on going clown tliiough the chain of peihcl engines, ue come to a point at which the last iiaction of Ihc heat has been com ei ted into woik That fixes the absolute zcio And we call Ihc steps by which we have come equal steps of tempciatuic, the steps being cletei mined by the consideiation that each engine in succession is to elo the same amount of woik out of Ihc icsuluc of heat icceivcd fiom the engine immediate^ before it in the sciies That fixes Ihc scale Moi cover the stops can be so taken, that the scale they give will agiee at two fixed points with the ordinary theimometiic scale, and will contain between those fixed points the same number of steps as the oidmary scale contains degiees Thus suppose the initial temperature, at the top of the chain, is that of the boiling point of watei, and that we have 373 engines in the chain. Then we find that it takes 100 steps to come -it THERMODYNAMICS [en. do Mil to the temperature of melting ice, and 273 moie steps 1 to complete the convcision of the remaining heat into work. This means that the uniform step of tempcialuic on, the thcimodjnamic scale is equal to the a\ eiage of the mtcivals called dcgiccs on any centigiade theimometei, when that average is taken between the freezing point and the boiling point (0 and 100), and that the absolute zeio is at a point 273 of such steps' ijeJow the freezing point But the llicimodynamic scale Mould agice from point to point vi Hi the indications of the thcrmomclci throughout: the whole of the scale oiuy if the theimomeler could use a peifect gas as ils expanding substance Even with h}diogen, which is ^ery ncaily a peifect gtiSj theie are slight cliveigenccs which \\eic mentioned in Ait 15. 43. Reversible Engine receiving Heat at Various Tempera- tures. In Camel's cycle it was Assumed that thcic was only one soiucc and one reccuei of heat All the heat thai was taken in was taken in at2\, all the heat that was rejected was, icjectcd atT 2 But an engine may take in heat in slaves, at moic tempciatuics than one, and may also icject heat in stages Wilh j eg aid to cvciy quantity of heat so taken in, the result still applies that I he greatest fraction of it that can be converted into woik is icpic- senlcd by I lie diffciencc between its tcmpeiatuics of rcccptioii and icjcclion, dnielcd by the absolute lempciatuic of icceplion And this is the 1'iaclion that will be coiucitid uilo work piovided the pioccsses within the engine aic icvci->iblc. Thus if Q z represents that paitol' Ihe whole suppb/of heal which is taken in at r J\ and Q, icpiescnts what is lakcn in at some olhci lempciatuic To, Qj at T 3 , and so on, and if T () be the tcmpeialuie at \\luch Ihc engine icjccls heal, UK whole \\oik done, il Lhc engine iti icversible, is ft (T, - T ) <2 2 (T, - T ) Q, (T, - T {] ) fr - ,,, -| ,-- -t- - ~,., r ci,c. J 1 ^2 J A We heic take, foi simplicity of statement, a single tcmpeialuie of rejection T A mechanically analogous machine \\ r ould l)c a gicat walei- Avhccl, working by gravity, and iccciving water mlo its buckets fiom leseivons at various levels, some of which aic lower than the top of the wheel. Let Jl/ a , M z and so on be the weights of water * Moie exactly 273 and a nation (Ait 10) FIRST PRINCIPLES 43 ccived at heights I l3 1 2 etc. above any datum level, and let 1 1 the height above the same datum level at which the water ives Ihe wheel If the wheel is peifecLly efficient (and hcie again c test of pa feet efficiency is re vei ability) the work done is MI (h ~ k) + M z (1 2 - 1 ) + IT, (1 3 - Z ) + etc. mipanng the two cases we sec that the quantity Q.JT l is the aloguc in the heat-engine of l/ x in the watei-wheel, Q 2 [T Z is <- analogue ol AI 2 , and so on. The amount of work which can got out of a <> i ven quantity of heat bjMeJttmg.ji_dg_w_n to an " ~l !~ '" ..',. '""V 1 ,, . '. ' '-..the (..!.i 'i . v ( , I. , 'i . L . i ; ! t to_ e piodnct ofJ2/TJby_ Lli^faJI^LtcinpciaJuiUL On the stiength of is analogy Zcunei has called the quantity Q/T the ''heat weight" a quantih ol heat Q obtainable at a tempciatuie T AnoLhei way ol' pulling Hie inaltci has a \\idei application. t the engine as beloie lake in quantities of heal icpicsented by ,^,Q,i'lc at^V/Vy'^andletil iqtct heal at T',T",T'" etc , 1 quantities u'jectcd l)cin<> ic'spcch\ el\ Q',Q",Q'"cic Tlan by 1 pnncipk thai in a ie\eisible cycle Hit heal u (ected is to the il laken in as the absolute lempeialuu ol ujeclion is to the -.olutc kinpeiahiu of icceplion, nc have Q' , ^2" , '" , _ i , ft , Q* , T' T" T" " 2', T 2 T 3 ^Q. m Avluch S = 0, en the simimahon is clTeeled nil lound Lhc ie\cisible e\clc llus suinmalion heal taken in is icekoned as ])osili\c and heat I'ck'd as negalivc If (he c\ele is not icversibk, the heat ic- Icd will be iclatively gicatci, and lliuefoic, foi a non-ieversible le, H (Q./T) will be a negative quantity ionic of the pioccsscs may be such that changes of tcmpeiatuie going on continuously while heal is being taken in or given out, 1 if so we cannot divide the icceplion 01 lejection of heat into nulcd number of steps, as has been clone above. But the equa- i may be adapted to the most geneial case by wntmg it sgration being performed round the whole cycle. This holds for any internally rcvei sible cycle It means that when 44 THERMODYNAMICS | < 1 1 a substance has passed through any senes of levoisible ohangis which cause it to leturn to its initial stale, the (pi.inlilns ol IK a I which it has taken in and given out aie so related lo UK lim- peratme of the substance at each stage as to make this mlcgi.il vanish foi the cycle as a whole If the cycle is not reveisible jdQ/T is a negative quantity, because the amount of IK a! icjected is relative^ laigei than when the cycle is icu-isihlc 44 Entropy. We have now to inlioduce an impoitanl Iheimo- dvnamic quantity vilnch serves many useful ])ui pose s The ttntio/>i/ of a substance is a function of its state which is most COIIM UK nll\ denned by icference to the heat taken in or gixen out x\h< n UK state of the substance undeigoes change in a icveisibk maniK i In any such change, the heat taken in or gnen out, divided b\ the absolute tempeiatuie of the substance, measmes I he- ehang< ol entropy Thus if a substance which is cilhtr e'X[)andmg n \ eisiblv or not expanding at all takes in heat 8^ wJien its le-mpi i.ilnx is T, its entropy mci eases by the amount 8Q/7 1 We shall see Ilia I UK entiopy of any substance in a definite slate is a delimit- <punlil\, which has the same value when the substance comes bae-k again lo the same state after undeigomg an> changes To gi\ e I h, e nl i ops a numerical value we must stait lion, some aibiliaiy p,.n,l uh, i, foi convenience of leckonmg, the entiopy is taken as /e,o \\ ,' aie conceined only with changes of entiopy and rons,,,,,, mix ,1 does not mattei, except foi convenience, what /c.o stale is H,,,,, toi the purpose of calculating the entiopy Starting then from any suitable zero,' each clement *Q ol UK heat taken in has to be divided by T, which ,s the absol,,| ( ,, ,, u . tuie of the substance when 8Q is being taken m Tin sum the enhopy of the subsUnoo, ,1,, ....... , ........ , , W, T'f " Se f StatC haS cc " ri ' 1K M- , ..... J la shall denote the ent.opy of v s,,bs llu ,. |, v ,/? , , , pe.ature chan s mo oont,no,,slj. wh.le henl ls I,, ', ,.,! , , change of enhopj- fiom any state lo an, o.iu,. s |.u,' 7," mevasibie MJttoT JfKINCIPLES "' 45 This definition of the entiopy of a substance as a quantity f 6 dO uch is to be measured by reckoning -^ while the substance a * sses by a reveisiblc process from any state a to any other state is consistent with the fact that the entiopy is a definite function the state of the substance, which means that it has only one ssible value so long as the substance is m the same state. To jve this we must show that the same value is obtained for the liopy no mallei what icvcisible operation be followed m passing 6 ,7/~l m one slate to I he othci in othei woids, thai f is the same a all icveisiblc opciations by which a substance might pass horn Ic a to state b Considci any two icveisible ways of passing in stale a lo state b If \\c suppose one of them to be icversed i two toi>ethci will foim a complete cycle foi which (by Ait 43) = He nee I r for one ol them iniibt be the same as t (l J. the oihci il is Ihcicfoica inattei oi'indiflcience, in the icckon- ,of entiopy, by \\haL ' path" 01 scqucnccoi cliangcs the substance >scs liom a to b pioudcd it be a icvcisiblc path staitmg fiorn y /cio slalc the leckonmfi of the enliopy in a gixui slate \\ill k r ays gi\e Llie same \alue, winch shows that the entiopy is iply a lunction of the actual state and docs not depend on 'vious condihons [I is chiefly because the entiopy of a substance is a definite iction of the stale, like the leinpuatuic, 01 the pic&smc, 01 the umc, 01 the internal eneigy, that the notion of entiopy it, nn- tant in cngmecung Ihcoiy The entiopv of a substance is idlly icckoned pei unit of rnasb, and numencal values of it koncd in this mannci aie given in tables of the pioperties of am and of the othei substances which aie ut,ed m heat-engines 1 refiigciatmg machines. But we may albO leckon the entiopy of a body as a whole when state ol the body is fully known, or the change of entiopy which >ody undciyocs as a whole when it takes in or gives out heat d we may also leckon the total entropy of a s} r stem of bodies adding togethei the entiopies of the seveial bodies that make the system 5. Conservation of Entropy m Carnot's Cycle. As a simple stration of the uses to which the idea of entropy may be put, * THERMODYNAMICS [cn considei the changes of enLiopy which a substance undeigocs when it is taken through Cainol's cycle (ArL. 32). All fom opera- tions are ie\ei'sible. In the fust, which is isoLhcimal expansion at T lt the entropy of the substance mci eases by the amount QJTj^ where Q a is the amount of heat taken in from the hot source. In the second operation no heat is taken in 01 given out and there is no change of cntiopy In the thud opciation a quantity of heat Qa ls rc]cctcd at T z the cntiopy of the substance accoidingly falls by the amount <2 2 /T 2 J- 11 tae fourth opeiation theic is again no transfer of heat and no change of cntrop}'. It is only in the fiisL and third operations that changes of enLiopy occui. Moicovcr they aie equal, foi Q l j2\=^ QJT 2 , which shows thai the substance has the same cntiopy as at fust, when it has icLuincd to the oiigmal state During the lii^t opeiation, while it was talcing in heat, its entropy rose fioni the initial value, which we may call (j> a , to a \aluc (j> b such that / / , /i in-, </>& = </> + QiM i Dining the thud operation, while the substance was rejecting heat, its entiopv t'cll again from cj) b to (/>, and ta - fa ~ W* Taking the C3'cle as a whole, the thermal equivalent of the woik done bv the substance is Qj Q.*, <i'id is accordingly equal to (T l - T 2 ] (^ ~ a ). Furlhci, the source of licat has losl ,m amount of entropy equal to Q. i jT i , and the receiver has gamed an equal amount of entrop\, namely Qj/jT, We may thercioie regard the reversible engine of Car not as a device winch Lransleis cntiopy Jiom tire hot source to I he cold receiver without altering the amount ol the cntiopy so tiansfeired The amount of heat altert> in Ihe process of UaiihCci, for an amount of heat Q t Q, disajjpeais, winch is the thermal equivalent of the work done, but the amount of entropy rn the system as a whole does noL change If, on the other hand, we had to do with an engine winch is not reversible, uorking between the same source and recerver, Q n would be relatrvclv larger, since less of the heat taken in is con- verted nrto work Hence Q 2 /T 2 would be greatei than Qj/T t and the amount of entropy would theiefore increase in the transfer 46. Entropy-Temperature Diagram for Carnot's Cycle. It is instructive to represent the changes of entropy rn a Carnot FIRST PRINCIPLES m Entropy Vis 5 cle by means of a diagram the two coordinates of which are the tropy of the woikmg substance and its tempeiatmc (fig. 5) ,e first operation (isotheimal expansion) is icpresented by ab, .tiaight line diawn at the level of temperatme T l . dining this jiation the cntiopy of the substance uses from </> a to <^ 6 This followed by adiabatic expansion be during which the tem- atuic falls but the entropy does not change Then it>othei- 1 compicssion cd at tempeia- c 7'o,duung which the entiopy , uinstolhcimtialvalue Finally aba I ic compression da com- ics the cycle j The aica of the closed figuie -5 d c d mcasuics (m heat units) the <u ik done duimg the cycle The (= a niabn mcasmcd to the base L ', which is the absolute zcio of ipciaturc, is the heat taken m n I he SOUK c The aiea indui is heat ic|ccled to Ihciccenci se (igmes aic icclanglcs ,11 [his is liue^\luitcvci be the \\oikmg substance Neither m 15 iioi lii-ii- is ,iii> assumption made as to that The diagram 5) applies to any engine going thiough the icvcisiblc cycle of not win UK i it use a gas (as in Ail 30) 01 anv othei substance 7 Entropy-Temperature Diagrams for a series of Rever- e Engines We may apply this hod ol lepiesenUilion to exhibit action ol the imaginary chain of isible engines which was used Lit 12 to establish a thermody- ue scale of temperature. artmg fiom any temperatme et a icversible engine take in at that temperature, and go i ugh the Cainot cycle of opera- > lepiesentcd by the icctangle For this purpose it takes m equivalent to mdbn and i ejects equivalent to mdcn. Let its ted heat pass on to the next 1-u X ilJii J.X1U.VJ LJ i IN AlU J. V,O [ CIJt ' engine of the series, which goes through the Carnot cycle dccf, and let the inteival of tempeiatuie rf/be so chosen as to make the ^ woik done by the second engine equal to the woik done by the / fiist. Fiom the geometry of the figme it is obvious that this requncs ilf to be equal to ad, so Lliat the area abed may be equal to the aiea dccf. Sumlail} in oidci that the woik clone by the thud engine should be the same, we must have fh -= df ad, and so on. Thus these mteivals constitute equal steps in a scale ol tempeiature which is based entirely on theimoclynannc considera- tions, the condition dctei mining the steps being simply this, that the same amount of woik shall be done by the heat as iL pusses j down through each step. < i 48. No change of Entropy in Adiabatic Processes It ibl- ^.(rf lows fiom the definition oi entiopy given m Ait 4.1 that uhcn a substance is expanded or compiessed in an adiabatic mannci (Ait 23) its entiopy docs not change An adiabalic hue is con- sequently a line of constant entiopy, 01, as it is sometimes called, an isentiopic line Just as isothermal lines can be distinguished by numbeis T a , T 2 etc denoting Lhe paiticulai tempeiatuie foi which each is chawn, so adiabatic lines can be distinguished by numbeis (j} l , 0, etc denoting the paiticulai value of the entiopy foi each We might accoidmgly define the enliopy of a substance as thai characteristic of the substance which does not change in adiabatic expansion 01 compiession, and this definition would be consistent with the method of leckonmg entiopy desciibed in Ait 44. Il is only in a icveisible piocess that the change of entiop\ of a sub- ^-fS? stance is to be deteimmed by lefeience to the heat it takes in 01 gives out. The definition of an adiabatic process (Ait. 23) excludes any piocess that is not reveisible 49 Change of Entropy m an Irreversible Operation It is impoitant in this connection to leahze that a substance may m- ciease its entiopy without having any heat communicated to it fiom outside When a substance expands in an it reversible manner, as by passing tlnough a thiottle- valve fiom a legion of high pies- suie to a region of lower pressuie, it gams entiopy Woik is then done by the substance on itself, in giving energy of motion to each portion as it passes through the valve, and this energy of motion v is futteied down into heat as the motion subsides thiough mteinal | fuction. The effect is like that produced by the communication I 49 some heat, though none is taken in from outside the substance. pansion through a, throttle- valve may be rcgaidcd as consisting two stages The fust stage is a moic 01 less achabatic expansion ling which the substance docs work in setting itsell in motion second stage is the loss of this motion and the consequent iciation within the substance itself of an equivalent amount heat Thcie is accoidmgly a gam of entropy, which occuis 3anse the process as a whole is not IC T , cisiblc \VL cannot dncctly apply the definition of entiopy given in I L4 to deteiminc the amount by which the entropy of a sub- nee changes in an nieveisible opeiation such as thiotthng. t when the linal state is known it is m gencial easy to calculate 1 cntiopy coiiespondmg to that state, by considering the amount which the cntiopy would ha\c changed if the substance had nc to thai state by a ie\eisible opeiation foi which $dQ]T amines the change ,Vhcn a substance has passed thiough any complete cycle of lations Us entiopv is Hie same at the end as at Lhe beginning, the oiiginal state has been rcstoiecl in all ict>pects This is tine in mcveisibk cycle as \vell as of a ic\eisible c^cle But toi an \usible c^clc jdQ/T does not vanish It has a ncgatixe value I l'3) and il docs not mcasinc change of entiop} , foi it is onK in inlcmalh ie\eisible action that the change ot cnhopyis Sum of the Entropies in a System It is mstiuctne to how the sum ol Hie cntiopies ol all paits ol a theimo- i.inuc system is allected \\hcn we include not onh the \\oikmg slancc but also the soincc ol heat and the sink 01 recen ei which heat is i ejected CoiiMdei a evchc action m \\hich the king substance takes in a quantity of heat Qi from a souice l\ and i ejects a quantity Q 2 to a sink at T z When the cycle omj)lefcd the source has lost cutropj to the amount QJT l tlic king substance has ictumcd to the initial state, and theicfoie neithei gamed noi lost cntiopy the sink luib gamed entiopy lie amount <2 2 /T 2 If the cycle is a reveisiblc one Q i /T 1 = Q 2 /T 2 , thcicloic the system taken as a whole, consisting of souice, stance and sink, has suffeied no change in the sum ol the topics of its paits. But if the cycle is not icversible the action ?ss efficient, Qa bears a l al ei piopoition to Qi ^nd Q 2 }T 2 is iter than Qx/^i Hence m an irieveisible action the sum of entiopies, of the system as a whole becomes increased This E T 50 llllkKMUlJ*r\AlUlA,S [LH. conclusion has a veiy wide application it is Line of any system of bodies in which Lheimal actions may occui U may be expiessed m geneial teims by saying that when a system under- goes any change, the sum of the entiopies of the bodies which take pait in the action lemains unalteied if the action is reversible, but becomes mci eased i( the action is not icvcisible No leal aclion is stnctly icveisible, and hence any ical action occui- img within a system ol bodies has the effect of mci easing the sum of the entiopies of the bodies which make up the system This is a statement, in teims of entiopy, of the pnnciple that in all actual transfoimations ol cneigy there is what Loid Kelvin called a universal tendency towaids the dissipation of eneigy ' Any system, left to itself, tends to change in such a manner as to inciejse the aggiegatc entropy, which is calculated by summing up the entiopies of all the paits The sum of the entiopies m any system, consideied as a whole, tends towaids a maximum, which would be i cached if all the eneigy of the system wcie to take the fonn of umfoimly dilluscd heat, and if this state were i cached no fuithei transfoimations would be possible Any action within the system, by increasing the aggiegate entiopy, bungs the system a step nearer to this state, and to that extent diminishes the availability ol the eneigy m the s}stcm foi finthei liansloimations. This is tine of any limited system Applied to the umvcise as a whole, the docLnnc suggests that it is m the condition o( a clock once wound up and now miming down As Clausius, lo whom the name cntiopy is due, lias icmaikcd, '"the enemy of the univeise is constant the cntiopy of the- univcisc tends towmds a maximum." ^W 1 ', 3"* An extiemc case of thcimodvnamic waste oca us in Ihc dncct conduction of a quantity of heat Q liom a hot pai t of Ihc sj stem, at 2\, to a coldci pait at T,, no woik bcmq done in the pio- cess The hot pait loses cntiopy by Ihc amount Q/^\ the cold part gams entiopy by the amount Q/T 2 , and as the lallei is gicatci thcic is an incicasc in the ay^u-galc qiianlily ol cntiO])y in the system as a \\holc. 51. Entropy-Temperature Diagrams. We shall now con- sidci, in a moie general mannei, diagiams in which the action of a substance is exhibited by showing the changes of its cntiopy in relation to its tempeiatnre Such a diagiam fonns an mteicstmg * AlalhemafiLCil and Physical Papeis, vol r, p 511 [j FIRST PRINCIPLES 51 md often useful alternative to the pressure-volume or indicator iiagram. One example, namely the entiopy-tempeiature diagram or a Carnot cycle, has aheacly been sketched in fig. 5. Let dcf> be the small change of entiopy which a substance under- ;oes when it takes in the small quantity of heat dQ at any tem- >eiaturc T, it being assumed that in the process the substance indeigoes only a leversible change of state. Then, by the definition f entiopy (Ait U), dQ ( '' < r = ~f ' thence Td(j) = dQ, nd JT<ty=Jd& 'ie mtegiation being perfoimed between any assigned limits. low if a cuive be diawn with T and </> for coordinates, jTf/<ji is the ica undci the cuive. Tins b} the above equation is equal to Jf/Q Inch is the whole amount of heat taken in while the substance isses Lluough the states which that poition of the cuive repre- nts Let ab, fig. 7, be any poition of the cuiv e oi and T. The ca of the cioss-hatched ship, whose bieadth is S<^> and height T, T8(f>, which is equal to 8Q, the heal taken in dining the small lauge 8cj} The whole area niabii 01 V/0 between the hmiLs a and b is !/ l e wliole heat taken in while the sub- mce changes in a icveisiblc mannci i 3111 the state icpicscntcd b) a lo ; c state lepicsented by b Sinn- | [y, in changing icvcisibly horn lie b lo state a by the line ba the bslancc icjccts an amount of heat n, t \n _ i 1 1 1 7 ElltUl/HJ I uch is mcasLiicd by the aiea nbam . . , . ., Fie 1 Enliopy Temperature ic base line ox conesponds to the fa Cm\o soli lie 7Cio ol tcmpciatmc. ^Vhcn an entiopy-tempeiatinc cur\c is diawn foi any complete j lc of changes it foims a closed figuie, since the substance icliuns its initial state. To find the area of the figuie we have to mte- tc thioughout the complete cycle, and piovided theie has been nre\ r cisiblc action Avithm the substance, being the heat taken in and Q 2 the heat rejected. But the eicncc between these is the heat conveited into work, hence 42 52 THERMODYNAMICS |cn. when the mtegiation extends round a complete cycle and W is expiessed in theimal units. Thus an entiopy-tempcratme diagram, so long as it lepresents changes of state all of which arc icvcrsiblc, but not otheiwise, has the impoitant piopeily in common with a piessuie^olume diagiam that bhe enclosed area measures Lhe woik done in a complete cycle But the entiopy-tempeiatuie diagiam has an adv.mlage no( possessed hy the piessuie-^olume diagiam, m that iL exhibits nol only the work done, but also the heat taken in and the heat i ejected, by means of aieas nuclei the cuives An illustiation of this has aheady been given in speaking of the Gamut cvcle (Ail <10), and others will be found in Chapter III. 52. Perfect Engine using Regenerator Besides the cycle <>P Cainot theie is (theoieticalty) one othci way in which an engine, can woik between a souice and icecivci so ab to make Ihe whole action icveisible, and theiebv tiansform into woik the gicalest possible piopoition of the heat that is supplied Suppose their is, as part ol the engine, a body (called a tk icycneialoi ") into \\lnch the woikmg substance can tcmporanh deposit heal, uhile. I ho substance falls mtempeiatuie fiom the upper limit I\ to the lowci limit T-,, and suppose lurthei that this is done in such ,1 in. unit i that the tiansfer of heat fiom the substance to the icguicialoi is leseisible. This condition implies that thue is to be no sensible chffeicncein temperatuie between the woikmg subsUme and UK matenal of the regcneiatoi at any place wheie they aic in I luini.il contact. Then when we wish the substance to pass back horn 7', toT^we mayre\eise this tiansfcr, and so icem u Ihe lic.il which uas deposited m the icgeneiatoi This alteinate stoimo ,,iul restoring of heat scives instead of adiabalic expansion ami com- piession to make the tempciatme of the woikmo siihsl.mou pass fiom 2\ to T z and fiom T, to f l\ icspeetivcly It t nables UK- inii- peiatuie of the substance to lall to 2\ bcfoieheal is K |(ckd lo llu icceiver, and to use to 2\ bcfoic heat is taken in liom UK soincc This idea is clue to Robcit Stilling, A \h o m ia-27 (Usi 4 nt'il an engine to give it effect Foi the pieCent jnnjwse il will snlluc lo descube the legeneiatoi as a passage (such as a group ol lubes) thiough which the woikmg fluid can Iravcl in cither dm-chon whose walls have a vciy laige capacity foi heat, so that the amoiinl alteinately given to 01 taken from them by the woikmg fluid causes no moie than an insensible rise 01 fall in their tempcralme ij FIRST PRINCIPLES 53 The lempciatuic of the walls at one end of the passage is T 1} and this falls continuously down to T z at the other end. When the working fluid at temperature 7\ enters the hot end and passes tlnoii>h, it comes out at the cold end at tcmpciatuvc T z , having stored in the walls of the regcncialoi a quantity of heat which it will pick up again when passing tlnough in the opposite dnection Dm mg tlve leluvn journey of I he working (hud through the ic- gcntraloi fioin (he cold to the hot end its temperatuic uses fiom T, to 7\ by pickino up the heal which was deposited when the woikino fluid passed through fiom the hot end to the cold The piocoss is shicllv rc\ cisible, or i at her would be so if the legcneiatoi had an unlimited capacity for heat, if no conduction of heat took place along Us walls fiom the hotter paits towaids the cold end, and it* Lhcre wcic no loss by conduction or radiation fiom its ex- ternal smfacc A icgcnciatoi satisfying these conditions is of conisc nn ideal impossible' to icalize in piacticc. 53- Stirling's Regenerative Air-Engme. Using an as the woiking substance , .ind <mplo\ing his icgenuatoi, Slnling made <iu enonu which, allowing loi piaclical impel lections, is tlie eaihesl example of a icxeisible engine Tlie c\ cle of opciations in Milling's engine nas substantial^ this (in describing it ue tieat an as a pufecl gas) (1) An, \vlueh h;id luui heated to T } b\ r passing tlnouqh the K'i>riKM at 01, ^v t ls allowed to o pand isollu i mall} lluoiioli .1 ialio ;, lakiiiH in lu.il fiom a Imiuee and laism^ a pislon Ilcnt taken in (pel 11) <>l an) - R r L\ lon i / (b\ Ait 2<S) (2) I'lu' dii was causal to pass thiouyli the icgcneiatoi fiom Ilic hot to the cold end, deposit ing heat and having its tempeia- hiu' loweied to T, without change of volume. Ilcat stoicd in legc'iieiuloi -- A",, ( r l\ T,) The piessuie of comse fell in propoi- lion lo Uu Jail in Icmpei.iluic (,'3) The air was then compi essecl isothcrmallv at T z , tlnough the same ratio ? to ils onginal volume, m contact with a icceivcr of heal. Heat rejected = JtT. 2 log c / (L) The air was again passed through the legenerator fiom the colcl to the hot end, taking up heat and having its tcmpcratuie laiscd Lo r l\ Heat restored by the regenerator = K v (2\ T 2 ). This completed the cycle THERMODYNAMICS [fir. The efficiency is n j log t i - RT 2 log, r __ Tj. - T z Volume "Fifi 8 Ideal Hulii nl 01 (lia"iiiin of An EIU-UUI \villi Ili'ji-iMim- afioi (Sdiliuj is RT, log, 7 T! Themdicatoi diagram of this action is shown in ho. 8 engine is impoitant, not as a piescnt- day heat-engine (though it has been icvived in small foims aftei a long mtenal of disuse), but because it is typical of the only mode, other than Camot's plan of achabatic expansion and adiabatic completion, bv which the action of a heat-engine can be made leveisiblc A modified foim of legeneiative en- gine was devised latei bv Encsson, who kept the piessure instead oi the volume constant while the woikmg substance passed through the legenciatoi, and so got an indicator diagiam made up ol frno isothennal lines and two lines of constant pussuu The entiopy-tempeiatiuc diagiam of a ie<>cuci;ili\ t of the type shown m fig. 9 The isotheimalopeiationof taking in heal air, is irpusui ltd l n ab, be is the cooling of the substance from TI to T z in its passage thiough the ic- geneiatoi, \\heie it deposits heat cd is the isotheimal i ejection of htat at T Z) and da is the icstoiation of heat by the icgcncialor while the substance passes thiough it in the opposite dnection, by which the 1cm- peiatnie of the substance is laised fiom T, to T! Assuming the action of the ic- geneiatoi to be ideally pcifect, be and ad aie piecisely similai cuncs wh,itc\ei IK- then foim The aiea of Ihe fio mc is then equal to the aiea of the iccluns-le which would icpiesent the oiclmary Cainot cycle (fig- 5) The equal areas pbcq and ndam measmc Ihe lunl slou-tl and lestoied by the regenciatoi When the woikmg substance is an and the icffcncmlivc chaum-s take place eithei undei constant volume, as in Stiilmg's engine, / (llll< (111 UllL'Uli) i ialoi Fit <f< Hd FIRST PRINCIPLES 55 under constant piessuie, as in Encsson's, the specific heat K nng treated as constant, ad and be are logantlimic ciuves with ie equation </> = ^ -f constant, being K v in Stirling's process and K$ in Encsson's. 54 Joule's Air-Engine. A type of an-engme was pioposed r Joule which, ibi seveial icasons, possesses much theoietical teiest. Imagine a chambci C (fig 10) full of air (temperature T 2 ), Inch is kcpL cold by circulating watei or otherwise, anothei ambei A healed by a furnace and full of hot an in a state of Fi 10 Jouln's npicbsion (lunpuatuic r l\), a compicssing c\hndci M by \\lnch may be pumped fiom C into A, and a \\oikmg c^hndei N in leh an liom A may be allowed to expand bcfoic pasMiiir back o tlic cold ch.imbci C We tJiall suppose the chambeis A and o be laigc, in oompaiison ^vith the volume of an that passes in l\ ibliokc, so that the picssuic in each of them may be taken as isibly constant The pump M takes in an fiom C, compi esses adiabatically until its press me becomeb equal to tlie piessuie A., and then, the \alve v being opened, dchveis it into A The licator diagiam foi this action on tlie pait ol the pump is the giam/c/w? m fig 11. While this is going on, the same quantity hot an fiom A is admitted to the cylmdei N, the valve u is then sed, and tlie air is allowed to expand adiabatically in N until piessuie falls to the prcssuie in the cold chamber C, During 56 THERMODYNAMICS [c'ir. the back stioke of N this an is disciplined into C The opcinlion of N is shown by the indicator cliayiJiin cbfj m h" 11. The nioa fdae measuies tlie woik spent in diivino I ho pump, Ilio aioa <// "is the woik done by the an in the woikino c-yhiuUi N The dil'feience, namely, the aiea tthid, is I ho nol amoiml of w.ik obtamedby canyinn the <>ivcn qnanlily of an Ihionidi ,-i eompkle cycle Heat is Lakcn in when the an has ils lempeial uio taisul a _Jt V Fig 11 Indicaloi dm<j;i am in Joiilo's An Kii'.'Mir on entenng the hoi thambci / Since llns happens al .1 pussnu which is bensibly conslanl, Ihe heal lakcn in whcieT B = T 1 , Ihe Lcmpeialme ol /, and '/' is llu l< nipi i.ihne i cached by adiabalic compulsion in Ihe pinup Similail\, Ihe heat lejcclccl Q - f (T T } whcie T d To, Lhc Icmperalme ol (', ami 7', is UK l< mpt lalnu i cached by admbalie expansion in N Sinei Ihe (\pansion ,uid compiession bolh lake place be I we en lliesami liiiiini.il pussiius, the lalio of expansion jmel ooinpiession is Ihes.une ( .dlini; il ?, we have 7' r j\ i, y-! T , T - 1 il * (Art 26), and hence also /Ti fit ni ri\ rii ri\ B o i It (i a ~ * it - >., , and - 1 a J il 'a J il Hence ' , /' _- r /' ; and the effieiency i] FIRST PRINCIPLES ^ This is less than the efficiency of a perfect engine working between /HP /7"T \ the same limits of temperature f l ~ 2 J because the heat is not taken in anel rejected at the extieme temperatuies The atmospheie may take the place of the chamber C that ib to bay, instead of having a cold chambci, with circulating watci to absoib the icjecteel heat, the engine may ehaw a ficsh supply at each stroke fiom the atmosphere, and chschaige into the atmospheie the air which has been expanded achabatically m N The entiopy-tempciature ehagiam foi this cycle ib chawn m fig 12, where the letters icfei to the same stages as m fig 11 After aehabatic ccmipicssion da, the air is heated m the hot chambci J, dnd the cuixe ab foi this piocess has the equation Fig 12 Enfciopy- toinpeiabiiici flia- /2 ' K pdT , . , > giam in Joulo's ~7j\ = ^ Jl ("'gi ~~ '0 l a) All EngllU.' Then aehabatic expansion ones the line hi, diul cil is anolhel logaiilhmic cinvc Joi the i ejection ol heat to C' by cooling unclii T T , ('& constant picssiuc The latio ^ , which is icpicscntcd by ^ in fio 11 anel by "~ m fig 12, shows the propoibion which the J nb volume of the pump J/ must bcai to the xolumo of the \\oikmg cylmdei N. The need ol a huge pump \\oulcl be a si nous diax\- backiu [>i act ice, foi it would not only make the engine bulky bul would cause <i iclatixely Luge pait of the net indicated \\oik (o be expended m oxeicoming inction xvilhm the engine itsell In the- oiigmal conception of this engine by Joule it uas in- tended that the heat should icath the woikmg an lliibugh llu walls of the hot chambci, fiom an extcinal soiuce. Hul mslrad of this we may have combustion ol luel going on w.thin the hot chamber itself, the combustion being kept up by the supply of ficsh an which comes in through the compicssing pump, and, ol couisc, by supplying fuel cilhei in a solid foim fiom time to him- tlnough a hopper, 01 in a gaseous 01 liquid form. In olhci woids, the engine may opeiatc as an intunal-comhuttion engine' Iiilcrnol-combusLion engines, essentially of the Joule type, e-m- ploy.ng solid fuel have been used on a small scale, but by far (he 58 THERMODYNAMICS fen r most impoitant development of the type is, to be found in tMu>iiK^ which T\ork by the explosion or binning of a mivhnc of .111 \\\ll\ combustible gas or the vapoin of n coiiibushblr liquid Tin thermodynamics of inteinal-combushon engines Mill lx nm- sidered in a later chajatei. We shall also see later (Chapter IV) thai i puclic.iMr n- Ingeratmg machine, using an for working suhsl.mci 1 , is ulil.iiiK d by making Joule's An -Engine \voik as a licaL-punip CHAPTER II PROPERTIES OF FLUIDS 55. States of Aggregation. In the previous cbaptei the only substances whose pioperties \veie discussed were imaginary ones, namely pcifect gases We have now to tieat of leal substances, such as steam, cai borne acid, 01 ammonia, which serve as work- ing substances in heat-engines 01 lefngeiating machines, and to examine then action and propcities in the light of theimodynamic pimciplcs. Any such substance may exist in thicc states oi allegation, solid, liquid and gaseous We aic mainly concei ned \\ ith the liquid and gaseous states, in eithei of Mhich Lhe substance is spoken of as a fluid The woikmg (hud in an engine is often a nuxtuie of the same substance in the h\o states of liquid and \ apoui , but in some stages of the action it may consist cntnelv of liquid, in otheis onlncly of \apoui The vapoui of a substance may be either sahualed 01 supei heated A vapom mixed \\ith its liquid, and in cquihbiium with it, must be satuiatcd Any attempt to heat the nnxtuic Avould icsult in moie of the liquid tinning into satmated vapoui But when a vapoui has been icmo\cd fiom its liquid it may be heated to any extent, thcieby becoming supeiheated. Thus when steam is fonncd in a boilci it is necessanly satuiated when the bubbles leave the watei, but it may be supeiheated on ibs way to the engine by passing thiough hot pipes which cause its tempciatme to use above Lhal of the boilei Any of the so-called peimanent gases, such as hydrogen, 01 oxygen or mtiogcn, is a supei heated vapour which can be i educed to the satuiatcd condition by gieatly lowering its tcmpeiatme At any one pressme the saturated vapour of a substance can have but one tempeiature the snpci heated vapour at the same pressuie may have any temperature higher than that In the change of state from solid to liquid, and again in the 60 THERMODYNAMICS [cir. change from liquid to vapour, heat is taken in, though the substance does not rise in temperatuie while the change is going on. The heat so taken in was said in the phiaseology of old wiitcis to be- come latent, and the name Latent Heal is still applied lo it. Thus the heat taken in by unit mass of a substance in passing, wilhoul change of piessiue, fiom the solid to the liquid slalc is called I he latent heat of the liquid, and the heat taken in by uiul mass m passing, without change of picssure, fiom Ihc slate of liquid to that of vapoui is called the latent heo,t of Ihc \npoui The latent heat of water is SO thermal units, which means that 11111 1 mass of ice takes in SO theimal units while it mclls, Ihc theiiual unit being one-hundredth pait of the quantity of heal icqiinul lo waim a unit mass of water from to 100 ccntigiadc The tempeiatuie at which ice melts is only vciy slightly affeclcd by the piessine (see Ail. 99), and the latent heal of walei is piacticallv the same at all ptessuies ordinal ily met with. 11 we assume the piessiue to be one atmosphere, ice melts at Hie tem- peratuie which is taken foi the lower fixed poml (0 C ) in giadua- ting a theimometcr (Ait 15) At a picssuie ot one atmospheie watei boils at Ihc Icmpcialme which is taken ior the uppci fixed point of the Ihc-Tinomelei (nameh 100 C ), and the latent heat of the vapoui is 539 y I hei mal units We shall see immediately that the tempeiatuie ;i( which UK change fiom liquid to vapoui occuis, and also the amount ol heal taken in dining the change, depends gieatly on the piessmc Al higher piessuies the tempeiatuie of boiling is lughei and the amount of latent heat is less In desciibing the piopeilies oi fluids it will save cncuniloeuhnu to speak usually of water, taking it as Upical ol the icsl II is itself of special mtciest to the cngmeci, being the \\oiking substance of the steam-engine, and the numciical 'tallies by winch its pro- peities are expicsscd aie bcttei known than those th.it u-lnle to other fluids But the definitions and thcimodyiunuc.il pimeiplcs which will be stated must be undcistood as applying lo fluids m geneial. We have now to considei m more detail some of Ihc points lhal ha\e been bnefly summaiizcd in this Aiticle 56 Formation of Steam under Constant Pressure. The piopcitics of steam, or of any othei vapour, are most conveniently stated by lefeiimg m the fhst instance to what happens when il is Ij Jt'JLtU.l'JiJLlJLiJto Ul< JbL/UJLUb 51 brined undo constant piessuie. This is substantially the piocess iv Inch occurs in the boiler of a steam-engine when the engine is at woik To fix the ideas we may suppose that the vessel in which > Learn is to be foimed is a long upiight cylinder fitted with a Cuctionless piston which may be loaded so that it exerts a constant prcssme on the fluid below Let theie be, to begin with, at the foot of the cylmdei a quantity of water (which foi convenience of state- ment we shall take as one unit of mass, 1 lb. say), and let the piston rest on the surface of the \\atei with a piessuie P. Let heat now be applied to the bottom of the cylmdei As heat enteis the water it produces the following effects in three stages- (1) The temperature of the watei uses until a ccitam tem- peiature I\ is leached, at which steam begins to be foimed The value of r l\ depends on the pailicular piesbine P which the piston excits Until the tcmpeialmc T^ is i cached there is nothing but Avater below the piston (2) Steam is loimed, moie heat being taken in The piston, which is supposed to continue to e\eit the same constant piessuie, uses No fuithcr incicasc of tempeiatuic occuis dining this stage which continues until all the watci is coin cited into steam Dining this sUigc the slcam which is foimed is saluiatecl The \olume which the piston encloses at the end of this stage the \olume, namely, of unit mass of satmated steam at piessuie P and con- sequently at tcmpciatmc r l\ will be denoted by V 6 (3) If moie heat be allowed to cnki aftci all the water has been coin cited into steam, the \olumc will incicasc and the tcm- pciatuic will use The slcam is then <,upei heated its tcmpciatme is above the Icmpciatmc of satiualion. 57. Saturated and Superheated Steam The difleience between satuialcel and supei heated steam may be c\picss>ed by saying that it walci (at the tcmpeiature oi the steam) be mixed with slcam, some of the watei will be cvapoiated if the steam is supciheatcel, but none if the steam is satuiatccl. Steam in contact with watci, and in thermal equilibimm Avith it, is necessarily satuiateel When satuiatccl its properties differ considerably, as a rule, from those of a peifcct gas, but when supei heated they appioach those of a peifcct gas moie and moie closely the faithei the piocess of superheating is earned, that is to say, the more the tempeiature is raised aboveT s ,the tempeiatuie of satiuation corre- sponding to the given pressuie P. 02 11-iJbJttlMUJJYlMAiVllLb [C'H. 58. Relation of Pressure to Temperature in Saturated Steam. The tempeiature T s at uhich steam is 1'oimcd under the conditions described in Ait. 56, which is called the tcmpeuiLiuc of saturation, depends on the \alue oi P. The i elation of pi ethnic to the tempeiatine of satuiation ^as deLcimincd \\ith great caie by Regnault, in a senes of classical expeiiinenls Lo which much of oui knowledge ot the pioperties of steam is due" 1 . Rcgnault's obsci- vations extended horn tempera tuies below the zeio oi the ccnti- giade scale, \Uieie the vapour whose piessme was measuicd was that given off by ice, up to 220 C. The pressmcs found by him, expiessed in millimeties of meicuiy, weie as follows, omitting those below C. as not iclcvant to steani-cngmc calculaLions . Pietwiio of animated sU'tini Tempeiatiue C m mm ot Moiomy 460 25 23 55 40 54 91 50 91 98 75 288 50 100 700 00 130 2030 160 4051 190 9420 220 17390 It will be seen from these iiguics that the picssinc of satuialul steam ns.es with the tempeiatine at a late which increases rapidly in the uppei regions oi the scale. Vanous empnical loimulas have been deviled to expiess the i elation ol pi ess, me Lo tcmpcuitme m satmatcd steam and to allow tables Lo be calculated in which inLci- mediate values aie shown When a table is available, houcvci, it it, moic comement to find Lhe piessme coiicspoiidino Lo n ivcn tempeidtme, 01 Lhe Lcmpciature coiicsjiondmg to a n iv un JHCSSIMC dnectly fiom it, eithei mtcipolaLing or chawing a poition oi Liu- cuive connectmo picssuie with tcmpciatmc when I he \ nines con- ceined lie between those that are stated m Lhe table. 59. Tables of the Properties of Steam. At the end of this book a numbei of Tables will be iound showing not only the ic- lation of the piessme to the temperature of saturation, but also various other pioperties oi steam which aie of use in enginccimg * Him. Imt F,a>ice, 1847, l XS1 An accouafc of KognauK/6 mothoda of espenment and a statement of lus results expressed m Bntish moiiamou will Lo found in Dixon's Tieatise on Neat (Dublin, 1S49). n] PROPERTIES OF FLUIDS 68 calculations. Tables of the propeiLies of steam have been calculated by Piofessoi Callendai, by methods which will be explained later, and have been published undei the title of The Callendai Steam Tables '. Fiom Callendai 's babies, which give the most authon- tative icsults now available, a selection has been made, with his pei mission, foi the piuposes of this book. The fig m cs which are given foi the piessure of saturated steam at vauoub temperatincs aie not taken directly fiom the measure- ments of Regnault, but aic infeired fiom a charactenstic equation which Callcndar has devised to expiess the i elation between pres- suic, volume and tempeiatuic within the woikmg lange The validity of that equation (within the range to which the tables apply) is demonstiated by the general agieement of the quantities calculated horn it with the best expcnmental icsults, in measure- ments not only of the pressuie at saturation but of other properties ol steam. The picssuics, howevei, which aie stated in these tables do agree \civ closely with the icsults of Regnault's obseivations quoted above It is only at the highest piessures that an appieci- ablc difference will be lound, and c\'en thcie it is not matenal In othci icspccls the Callendai tables will be found to diffei somewhat widely fiom the eailiei tables of such authoiities as Rankme | 01 Zcuner |, which have been accepted as standaids and copied into many text-books When these were calculated the only a\ ailable data of value wcie those kunished by the expenments of Reimault But moie iccent icsearches have supplied additional data which in some paiticulais modify his, and it is now clcai that Regnault's hgtiics lequue icusion and m some cases consideiable anicndnicnl The vanous piopcities of ste.un, 01 of any othei vapom, aie linked together in such a mannei that the lelations bctucui lliein must satisfy ccilam theimocl^ namic equations This alfoids a tcsl of consistency, and in the light of such investigations the (iguies given in the old tables aie now known to be not even mutually consistent Callendai 's tables give a set of values that aic consistent amongst themselves and aie also m good agree- ment Avilli I he most tiustwoilhy cxpeiimenial lesults Further ic- scaiches may in Lime lead to a still closei adjustment of the figuics to the u'sulls of observation, but Callendai's \alucs lor the various * London, Edward Arnold, 1916 Students should obtain a copy of those Tables, which contain fulloi pailiculara than arc quoted hpre f- Rankiue, A Hatmal of the Steam Engine and other Prime Movers f Zouner, Techmsohc ThermodynamiL, vol n (Tians by J F. Klein, 1907 ) [CII. quantities may be accepted not only as mutualty consistent, fiom the Lliermoclynamic point of view, but as ceitamly collect enough for the purposes of the engmcei. 60. Relation of Pressure to Volume in Saturated Steam. Among the quantities shown in the tablet, is the volume F 4 , in cubic feet per lb., of saturated steam at vaiious temper atuies and at vaiious pressuies The volume of a given quantily ot saturated steam at any assigned temperatme or pressure is a quantity difficult to measiue by diiect expeiimcnL, and the volumes which aie given m steam tables aie generally mfcned from the results of expenments on other piopeities which can be moie easily measured. Successful mcasuiemcnts of \ olume ha\ eliowe~\ er been caincd out ' and the icsults aie in geneial agieement with the figures stated in these tables. The iclation of P to V s in sat ma ted steam is appioximately expiessed by an empnical formula PF s lu = constant. With P in pounds pei sq inch and V a in cubic feet pci lb this gives PV a l * =490. WiLhP in kilogiammcs pei squaicccntimetie and V s in cubic meties pci kilogiammc, it becomes This ibiniula applies, well liom a piessuie of say 1 pound pci squaie inch up to SOU pounds pei scmaic inch Within these Jimils it gi\ es "\alues which agiec to one pait in a thousand with those I in the tables. ^^ The student will find it useful to chaw cuives, with the data ? , of the tables, showing the i elation bet\vcen the picssure and the J temperature oi sat mated steam, anel also the iclation of piessuie to ,' volume, especially within the lange usual in steam-engine piacticc lie will obsen e that the late of change of piessuie with icspect to , change ol tempeiatuie mcicases lapielly as the tcmpeiatmc uses, * and lience that in the uppci pait of the range a veiy small ele\ ation of temporal ure in a boilei is necessnnty associated with ti large mciement of press me The piessuie shown by a piessure-gauge on a boiler is the excess ',* of piessuie in the boilei above the press me of the atmosphcic. * See especially Knoblauch, R Lmdo and H Klebe, lUittedungtn ilbcr For- sUuinijsarbeiteii hcraitsgeyeben vom V&em dadsclicr Ingemcurc, Heft 21, 1905 r] PROPERTIES OF FLUIDS 65 xonsequently the true or "absolute" piessure in the boiler is to >e found by adding, to the leading of a correct gauge, the piessure /Inch coriesponds to the height of the barometer at the time, this 3 generally about id- 7 pounds per square inch or 1-033 kilogrammes '6r square centunetie. 61. Boiling and Evaporation. The familiar case of water oiling in a kettle 01 other open vessel is only a special example of lie foimationof steam under constant pressuie. There the constant icssuie is that of the atmospheie, and consequently the temper a- iire at which the water boils is about 100 C ^ Watei in the open evapot cites slowl}*" at any tempeiature lower lian that at which it boils. Though the presume of the vapour so nmed is lowei than that of the atmospheie and may be veiy inch lower the vapour is able to escape from the suiface by if fusion the atmospheie is not displaced and the piessuie on the irface of the water is still that of the an As the tempeiature of r atei in the open is laised this slow evapoiation fiom the suiface ccomes moie rapid, but it is only when the tempeiatuie teaches ic valucwhich coiicsj)onds(loi satuiated steam) to the gi\ en atmo- )licnc pressure Lhat the watei boils the \apom is then foimecl in ubbles at the piessuie of the atmospheie, and it escapes not by iffusion but by displacing the supei incumbent an 62. Mixture of Vapour with other Gases: Dalton's Prm- iple In what has been said about the iclation oi picsbiue id volume to tempeiatuie in the satmaled stale, it has been ruined Lhat in the pioccss ol formation Lheie is simplv a mixture 1 the liquid with its vapour, no olhci substance being present. his is substantially tuio in a steam boilei 01 in the e\apoiator of icfiigciating machine. But the case is diffeient when the A apoui is to mix with anothei gas or gases. A pnnciple discovered by altou Lhen applies, that the piessuie in any closed space con- iininq a mixture of two or moie gases at any given tempeiatuie vciy approximately equal to tlie sum ol the prcssines which each 'the gases would exert separately if the others weie absent, that to say if each of the gases (at the same temperatuic) alone cupicd the whole space These pressure^, which aie added igcther to make up the actual pressure, aie called "partial * Water in the open boils at 100 C when the atmospheno pleasure has its mdard value, which coriesponds to a barometer reading (collected to 0C ) of mm at sea level in latitude 45, or 759 G mm in London (see Art 12) E. T. ^ 06 THERMODYNAMlLb t i.i, piessmes." An impoitant instance of the application of Dalton's pimciple is consideied in the next article. 63. Evaporation into a space containing Air* Saturation of the Atmosphere with Water- Vapour When watei cv.ipoi- ates in a closed space containing an, the piocess qocs on unlil ;i definite amount of it has become mixed, as \apoui willi Hie an aheady theie When this has happened and a state of e<|iiilil)iumi is leached, the an is said to be satuiatul Mith walci -\apom. The amount of watei-vapoin that a t>neii volume of <ni will lake up in tins way depend^ upon the tenipciatinc it is \eiv ncaih UK- s.inu amount as would be lequiied to fill the same space wilh salmalc d steam at that tempeiatme il the an weie not puseut J?y ])alloii's pimciple the piessme of the mixed oases, namely the an and UK watei-vapom mixed with the an, is ven> ncailv the same as UK sum of the piessaiies which each would excit sepaiatcly LluL is to say the piessuie in the given space aftei the wiLci-\ tiponi has been foimed is gieatei than the pies>suic whuh Hie an would c\eil in that space, if the watei-vapoui wcie not thcie, by an amouiil which is neailv equal to the piessuie of satinated sleam al Ihe tempeiatme of the mixtuie It is appioximatelv hue lo say Ihal each of the constituents of the mixed atmospheie in UK closed space behaves as if it occupied the \\holevolume, and conlubulcs to the piebsuie just as, if the othei constituent weie ahsenl This is veiyneaily accurate at 01 dmaiypicssuics It becomes Jess aec uiali vxhen thepiehsuie is high the amount of watei^apom zeqmud lo satuiate the atmosphcie is then somewhat less than Ihe iiilr uonld lequne. As an example, suppose an at 25 C (77 Fah ) lo be sal male d with uatei -vapour At that tempeiatme one Ib of salmaled sleam would (by the Tables) occupy 092-4 cubic feet, and lluieloiv O.K cubic foot weighs 0-001 JJ, Ib Consequently each cubic fool ( I he an takes up 000114 Ib of watci-vapoin m icachmj. UK sl.iU- of satmation at that tempeiatme And since the eoucspomlm.. piessuie of waters apoui is 46 pound pei sq inch, Ihr pn ssiur in an enclosed space containing this moisi an is qic.>lci by K, pound per sq. inch than it would be if Uie walei-vapo.u wcu- removed and the diy an alone weie left lo fill the same spae ( al the same tempeiatme In othei wouls 16 pound per sq meh is the "paitial piessme" of the watei-vapom piesent in Lhe an undei the assumed conditions. When the amount of watei- vapour present in air is less than tough to cause safciuation the watei -vapour is held in a supei- ;ated state. If the tcmpeiature of the mixtuie be lowered, a point reached at which the air becomes saluiated, and any further wcimg of the tempeiatuie causes some of the vapour to be de- bited as liquid on the Avails of the containing vessel, 01 on any u Licles of dust that may be piesent. Any solid paiticles will sei ve nuclei for condensation. The water condensed on such nuclei ims a mist of minute diops which fall so slowly that they seem to i held in suspension The tempeiatuie at which watei begins to 1 deposited fiom moist an is called the dew-point Condensation ' some of the watei contained in an will also occur on any cold irface (colder than the dew-point) uith which the air comes in mtact this icsults fiom local cooling of the an close to the suiface question. Thus in a refngeiatmg plant with pipes that convey a -imd coldci than the fleering point thiough the waim atmospheie ' the engme-ioom, a coating of ice forms lound the pipes Foi the me icason an effective \\ay to diy an is to make it cold and chain \ ay the watei condensed in the piocess, at the lowest tempeiatuie ic an icmains satmated, but the amount of watei lequned to tmatc it at a low tempeiatuie is vcivsmall,anrl uhen it is allowed become waim again without taking up moie water it will be fai um sal mation 64 Heat required for the Formation of Steam under Dnstant Pressure 1 Heat of the Liquid and Latent Heat ctuin now to the imagmaiy expenmeni of Ait 56, \\heie steam fonncd under the constant piessure of a loaded piston, nothing it watei 01 watei-vapom being piesent and enqiine what nount of heat has to be supplied in each stage of the opciation i i\\cfif,t stage the substance is wholly in the condition of A\atei Inch is being heated fiom the initial tempeiature to T 6 , the mpcialuic at uhich Ihc second stage begins. Dining this fiist age the heat taken in (per Ib of the watei) is appioximatcly equal one thcimal unit foi each dcgiee by which the tempeiature of the aler uses It would be exactly equal to that if the specific .heat watei wci e constant and equal to unity, but this is not the case, t about 30 C the specific heat of water is less than unit}', it isscs a minimum value thereabouts of 9967, and then increases, ^coming appreciably greater than unity at such temperatmes as e found in steam boileis Thus for instance to heat 1 Ib of water 68 1 lUiltiVLUJJ X IN AUI A^- I from C. to 80 C. icqimes 79 9 theimal units instead of 80 On the othei hand, to heat it fiom C. to 200 C , undei 1 a piessuic sufficient to pi event steam fiom foiming, requues ncaily 20.3 2 thermal units instead of 200. These figuies will indicate how 1'tu it is legitimate to estimate the heat taken in dining llic iirst sliiL>c as one unit pei clcgiee Moie accuiatc values oi the heal ol Ihc- liquid, that is to sav the heat taken in dining the fiist stn^e', can i be found by means of the Steam Tables (see Ait. 09). Dining this first stage, \vhile the substance is still liquid, ncai ly .ill the heat that is taken in goes to meicase the stock of uitei nal em i gy . Theie is scaicely any exteinal \\oik done, foi the \olume is only slightly inci eased. Thus m heating watei horn C to 200" ('. i (undei apiessuie of 225 2-1 pounds pci s<] inch) the volume ol I he- water changes horn 0160 cubic ft. pel Ib to 0185. The e\le-i nal *y* woik done dining this, heating is thcieioie 225 21 x 111, x 0025 or SI foot-pounds This is equivalent to baiely 00 theimal mill, and is negligible in companion wilh the 203 2 umls of heal llul aic taken m j In the second stage, the liquid changes mLo satmale-el sle.im wilh- I out change of tempeialuic The heat lhat is laken in elm ing ih ls stage constitutes what is called the Latent Ifcat ol [lie- \.ipom We shall denote it bv L Values of the latent jic.it of ssihnalnl steam are gi\cn in the tables Foi steam foimccl umln ,1 pie-ssim- of one atmospheie (salutation tempciatiiic 100 C ) Ihelalc-nl lu.il is 539 6 withlo\\ei piessuies of formation il is gievilci, jud \\ilh lughei piessmes it is less At the end ol the second sl,ii>c Ibi sub- stance contains no liquid, it is spoken of as diy s.iluiale-d sleam at any eailiei point, \\ hen the substance consists p,n lly ol s.il male d steam and paitly ol watei, it may be spoken of as we I sk-ain The latent heat of a \apom may be tledix-d as I he amounl ol heat which is taken in bv unit mass ol Ihe liquid while il .ill e li.ui'x -, into satuiated \apour uncle i constant piessine, Ihe liquid b.mun been pieviouslv heated up to the tempi-mime al Jmli flu \.ipoin is fonned A consideiable pail of the heat taken m dm ing M,is pmerss , s spent in doing external woik, since Ihe substance expands ng.uusl the constant piessme P. It is only the icmamder ol the so-calh-d latent heat L that can be said to remain m the ilmd and (<> con- stitute an addition to its stock of internal cncigv The amounl f spent m doing exteinal work during the second stage- is \ AP (V V \ - rj - i \' i r ; Jj n] PROPERTIES OF FLUIDS 69 whcie V s is Lhe volume of the saturated vapour and V w is the volume of the liquid at the same temperature and piessme, A being the factoi for coiu citing units of work into theimal units. The excess of L above tins quantity measmes the amount by which the internal eneigy inci eases dming the second stage Thus foi instance when water at 200 C and a pressme of 225 24 pounds pei sq inch is converted into steam, of the 467 41 theimal units taken in, 47 61 units aie spent in doing external woik "" and 419 8 units go to mciease the stock of internal energy. 65. Total External Work done. In the two stages together the whole amount of exteinal woik done is to be found by taking the whole increase of volume and multiplying it by the piessure. If we assume that the watci is onginally at C. its volume may be taken as 01CO In converting water fiom C to saturated steam at 200 C. undei constant piessme the exteinal woik done is found thus to be equivalent to 47 67 theimal units this is 06 units more than the external woik of the second stage, foi it in- cludes the small amount ahcad\ icfmcd to as ha\mg been done dming the fust slugc The nholc mciease of mtcinnl encigj , fiom water at C lo satinatcd steam at an\ tcmpcialiiic, is equal to the \\l\o\c amount of heat tal en in, less the eqimalent of the cxlunal \\oik done This in fact is onh a p.uticulai example of the gcncial pnnciplc stated in Ait 9, that \\hcn any substance ex- pands in any mannei. taking in heat anddoin" \\oik, the heat taken in is equal to the \\oik clone phis the mciease of internal cncigy In Ihc case hc'ie considcicd the aclion is oom<i on nuclei constant picssuio, but the statement applies to any change of state whatevei. 66 Internal Energy of a Fluid No mattci what changes a substance may undcigo, its inteinal eneigy \\ill ictuin to the same \aluc when the substance ictmns to the same condition in all u'specls In othci woids the inteinal cncigy is a function of the actual slate of the substance and is independent of the way in which that slalc has been icachcd Thus the inteinal eneigy of 1 11) of sal ura led steam at a paiticulai piessme is a definite quantity which is the some whcthci the steam has been foimed by boiling under constant piessme or in any othei mannei. Steam foimed in a closed vessel of constant volume, foi example, would have the same internal energy as steam at the same piessure but formed nuclei conditions of constant piessme, though the amountof i * The volume of the watei is 0185 cubic ft and of the steam 2 0738 cubic ft The value of AP ( F, - F, t ,) is therefore 47 61 70 THERMODYNAMICS | t -ji. heat taken in dm ing its fomiation would be diffeicnl, fornoexlriual work is done in the piocess of foimation in a closed \rsscl <>l con- stant volume In that case the hc.it Liken in would hi equal to the increase of internal eneigy We have no means of measuiing the total stock of inlciual eneigy in a substance, and can deal only with chants m llu- stock But by taking some arbitiaiy staiting poml as ,i '/cio fioin \\lnch the internal eneigy E is icckoncd we can r>i\e/i'a nnineiical \ alue foi any other state of the substance. That value ically cxpicsscs the diffeience fiom the internal eneigy m the zcio slate The usual convention is to wnte E = when the substance is in llu liquid condition at a tempeiatuie of C, and al apicssmc equal lo I he vapoui-piessure coirespon cling to that tempcialiue \Ve may call this, foi bievity, the zcio state of the substance Following this convention we take E = toi waloi al C The \alue of E for satmated watci-vapom at C will llu n be 5<> |, 'Jl thermal units (see Tablet, in Appendix) Thai this apices \\ il h <>[ lu i figuies in the tables will be seen by consiclcuri" I he couvcrsuin ol watei at C to steam alfprC. uiulci constaul pussinc Tbc <ml\ heattaken in is L, which is 59 J 27 units, and ol tins I hcc \l CM n,i I woi k AP(F b - V K ) icpiesenU 30 06 units- the dilfciencr niusiiirs K Values of E foi saturated steam at vnnoiis li-nijjtialiiirs aie given 111 the tables It will be seen lhat they mcirasc slow^ \\-ilh the tempeiatuie. 67 The "Total Heat " of a Fluid We co.nc now ID i function of the state of any substance, a fuiu.-l.on which is ol vn v gieat use in theimodynamic calculations. II Js frc-iH-i.illy callc d Ihc " Total Heat" and is repicscnled ' by I he IctkT/ The "total heat" / is defined lor any slate of (he subslancr ), v the equation = That is to say / is equal to the sum of [he intern,,] em^y ,,,, ( | || 1( exteinal woik which would be done if ll lc subsL.ncr could !,< imagined to staiL fiom no N olume al all and lo exp.uul lo Us acl,r,l volume under a constant pressuu- equal lo Us aclual t so the actual state, I is also a funclion of llu aclual sl,,|, Us value , s ndependent of how the state has been ,,,! r,, slnun <u exam^e, the heat taken m during fouuahon depends on how I he * Callendar , h, s Table, u a os // to iopioBO.il HUH funohon J u WI1W tlf Ul(1 Jilofc rl PROPERTIES OF FLUIDS 71 .team is formed, but the "total heat" / depends only on the final 'ondition The total heat can be calculated for any condition of a .ubbtance, whether in the slate of liquid or of saturated 01 super- icatcd vapoui. It is measiued in thermal units per Ib. Values of he total heat of saturated steam and also of water under satma- 1011 pressure at vanous tempeiatmes aie given in the tables. The olal heat of steam increases piogicssively with the temperature, cither moie lapidly than does the internal energy. It follows fiom the definition of I that in the zeio state of any .ubstancc, at which E is reckoned to be zeio, / is not equal to zero jut to a small positive quantity depending on the volume of the iquid and its piessme at that state Since E is then zeio I is equal AP V , whcicP is the picssure at the zeio state, namely the /apoiu-prcssuie at C , and F is the volume of the liquid at C. ind piessme P Foi water this quantity AP Q V is quite negligible, amounting as it docs to 000146 theimal unit For caibonic icid it is about 1 theimal unit, foi ammonia and sulpliiuous acid t is much less 68 Change of the Total Heat during Heating under Con- stant Pressure An mipoitant propcrt) ot the function I is that when an) substance is heated undei constant piessme the change jf / is equal to the amount of heat taken in To piove this, let Q be the amount of heat taken in while the substance expands undei joustant piessme P fiom a state in which the volume is F L and the inleinal eucigy is E-^ to anothei state in which the volume is V 2 and the intcinal cnciy) is E 2 Then the amount of external woik done is P (V VJ and, by the conscivation of eneigy, Q = E,-E i + AP(V.-V^ which may be wnttcn Q = Ei + APV, - (E : + APVJ 01 Q. = I? ~ AJ where Z^is the total heat in the fust state and / 2 is the total heat m the second state 69. Application to Steam formed under Constant Pressure, from Water at C. The above pioposition applies to cveiy stage of the imaginary expcnmcnt of Ait. 50 Refemng to that expeii- mcut, assume that lo begin with there is undei the piston 1 Ib of water dt C. and at the pressure P at which steam is to be formed By definition of the Lotal heat, I = E + APV, 72 THERMODYNAMICS [cli. E at the beginning may be taken as zeio" 1 . Hence the value of J foi the water at C maybe taken as APF Q . whcie F is the volume of 1 Ib of watei at C and P is the piessuie at which steam is lo be foimed. At the end of the fust stage \vheie I w lepiesents the value of/ toi water at the tcmpcinluie at which steam is about to form [ . When values of /, aie known I his allows Q 15 the heat taken m dining the fust stage, to bi inoic accmately calculated than b}*- the lough method of A) I G-i Values of I w are included in the steam tables Dming the second stage an amount of heat equal io L is la km in at constant piessuie, and the total heat changes fiom J w to I a , where I s is the total heat of saturated steam. Hence /. = L + I w = L + Q^ + APF The sum L + Qj is the whole heat of formation, in I he experi- ment of Ait. 56. Thus the "total heat" of steam is equal lo the heat of foimation uudei constant piessuie, plit^ ,i small quantity \\hichis the theimal equivalent of the woik Uuil would |H> done in lifting the piston fai enough to admit the onginal \oluiiu' of the watei The quantity APV Q foims a \uy small pail of UK ' total heat" it is only 37 theimal unit when the tempciahue of foimation is 200 C and it is much less at lowei tern pei.i Lines These lemaiks and the following tabula i scheme \\\\\ st'i\t lo show how the total heat of satmatcd steam (01 other vapom) is i elated to the heat of formation under constant piessme Bui Hie student should accustom himself to think of the lolal heal wilhoul lefeience to any piocess of foimation, as a piopcrly which a substance possesses in its actual state n piopeily \\luc-h is pisl .is simply a function of the state as is the tempt ial me, 01 (he pic ssinc, 01 the \olume, 01 the internal cncigv, or the cnlmpy, which we shall ha\e to considei piesentlvj. * The convention of Ait 60 makes 0=0 foi wiloi alOC and piosNino J>, lie,,, andpreasuioP.whioInslughoi 1( , H si,,( does not cause the internal energy of wntoi atOC to dilku upptwiahly fiom /.mo In C'allendar's Tables tins quantity / JH win ten 7; T The function here caUed tbo total hoat /, namely K , JPK, W1H mlllfl( | llc . (M i ,, y Willard Gibbs (T,a, ta oj Ike Connecticut Academy, vol nr, Collated Xnu^tu Papers, rol i, page 92), and was fii st called Iho " Total 1 loat by C.llni.lat ( May 19ui vol v, p 50) Its gtoal iniporLanco in tcchmcal tliOL.iiodynain.CH emphas^d by Molber, who omplo 3 ed it m chaits foi ovl,,bilmg tho pu^u of steam and other .ubstanccs Tho uso of such ohaita will bo dwonbocl lutoi I] PROPERTIES OF FLUIDS Total Heat, / E 73 APV ternal eneigy inucd by the toi in being id fiom C , Intci rial encigy acqimed during change of state from watei to steam, L-AP(V a -V w ) External woik done during change of state from water to steam, External work clone while the watei is being heated, APV. Heat taken in during second stage, L Heat taken in during first stage, Q 1 70 Total Heat of a mixture of Liquid and its Saturated /apour It follows from Ait 68 that while a liquid is being con- r ci I eel into \ apour, nuclei constant pi assure, the total heat 7 inci eases 11 piopoihon to the amount of vapoui that is t'oimed At am nleinieduilc stage in the piocess, if we call q the fi action that is apou/cd and 1 - q the haction that is still liquid, the total heat >f Ihc niixtine is T ql s + (1 -(/)/, vhieh may be uiillcn , !, + & Similaih, Mhilc a \apoui is bemo condensed under constant Hcssmc, / becomes less by an amount mcasmecl by the heat i\ci\ out, \\lnch is ])ioj)oitional, at am mtei mediate stage, to he fi action then condensed 71 Total Heat of Superheated Vapour When steam, 01 ny othci \apoui, becomes superheated (as in the thud stage of the \peiiincnL of Ail 50) b^ continuing the heating piocess undei onslanl piessuic after 1he satuialcd condition has been leached, lie value of I becomes inci cased above the "<; alue 1^, by an amount qua! to the heat so taken in We might find the total heat of upei heal eel steam by calculating the supplemental y amount taken n eluung I he piocess of supeiheatmg, piovided we knew the specific teat of the vapoui dining the process of heating it, undci constant nessiiie, fiom its Lempeialuie of satin a I ion to its actual tempera- me Bui this specific heat is not a constant it diminishes slowly s the temperatuie uses, and it is greater at high pressures than at ow pressuics A better way of finding the total heat in supeiheateel team is to use an equation, devised by Callendar, which gives the 74 THERMODYNAMICS [vu. total heat of the supei heated vapour directly for any condilion ol temperatuie and presume, without iclcicnce Lo the mode ol lonna- tion Tins will be descubed in a later cliaptci, and a selection of numeiical values will be found in the tables luom Ihcin I he heal taken in dining supeiheating at constant piessuie may be Jound as I' !, \\heie /' is the total heat m the supcihcatcd slale and I s the total heat in the satmated state at the same piessuie. In engmeeimg piactice, the supeiheating ol steam is generally earned out at constant piessiuc the steam on leaving the boiler passes thiough a gioup of tubes foiming a superhcaLci, kc])l hoi by the furnace gases, and while taking up heat fioni these tubes ils piessuie leniams equal (01 neaily equal) to that in the boiler Super- heating is laiely earned fmthei than 100 C and not oflcn so fai. 72. Constancy of the Total Heat in a Throttling Process. An important piopeity of the function /, in any substance, is thai it does not change when the substance passes thiough a valve or othei constncted opening, such as the porous plug ol' the Joule - Thomson expeiimcnt mentioned in Art 19, by which it becomes thtottled 01 'wnc-chawn" so that its piessuie chops A piacheal instance of this kind of action occius when steam passes lluouoli a paitially closed 01 ifice 01 "leducirig valve." Eddies aic louncd in the fluid as it uishes thiough the constncted opcnin 5 cind I hcciu-iyy expended in foiming them is flittered down intohcataslheysnbsidc is U To pio\ e that / is constanl in such nn opera I ion we shall consider what happens while a unit quantity of the subsUmcc passes Duonuh a constncted opening (as in fig 13), and, lo make the mallei clem, imagine this unit quantity to be scpaiated horn I he iesl ol (he substance by two fnctionless pistons, one of which ( /) slides in NIC pipe that leads to the constiiction and Hit ollui (ft) slides in (he pipe that leads away fiom it On one suit , as Die substanee comes up, let its piessuie be P l5 volume Fj and mtmuil cneigv J^ () the othci side, aftci passing the constiiction, lei its piessmc b< /', volume F s and internal eneigy E z As cacli po, l lO n appioael.( s I \u constiiction, ^oikis done upon it bv Die subslance behind pushing I PROPERTIES OF FLUIDS 75 the imaginary piston A, and the amount of that work done hilc unit quantity is passing is P 1 V l After each poition has issed the constnction it does work upon the substance in front T pushing out the imaginary piston B, and the amount of that 3rk is P Z V 2 for tlic whole unit quantity. Any excess of the work me by the substance on piston B over the woik done upon it by ston A must be supplied by a reduction in its stock of mteinal ergy. Hence sm which #> + AP Z V 2 = Ej + APJ\, ins the total heat does not change in consequence of the thrott- ig. The imaginary pistons Aveie introduced only to make the asoning more intelligible, the argument holds good whethei they e there or not It applies to any fluid, and to any action in A\ Inch eic is a fnctional fall o( pressure We might accoidingh describe thequantiU J as that piopeiiy of a b&tanct' which doa> not change in a thiotthng pjoccss l . 73 Entropy of a Fluid. In reckoning- the entiopy of a fluid _' foil on the sume comcntion as in reckoning internal eneigy the itropv of lire liquid at C is taken as zero Consider, as before, pioccss in which the liquid is first heated undci constant pressure id then vapoiized at that pressure During the heating of the [ind fiom an initial tenrpeiature T n to am tempeiatuie T (on the >solulo scale) the entiopy incieascs b} the amount f r dQ T adT \T~T ~ T a T ' heie a is the specific heat at constant piessure. If a could be treated as constant this Avould give on integration a (log. T- log. T ) In the case of Avatcr cr is not far horn constant and equal to unity, cncc a rough value of the entropy of water <, at any temperature is given by the expicssion loo t T - loo, 278 * It is assumed that no heat is taken in or given out and also that the velocity the pipes is so small that no account need bo taken of any chftcience m the kinetic oiffy of tho slion.m m the pipes befoie and after passing the constnction, once D eddies have subsided If the stream has acquired an appreciable amount of ictjc eneigy aitoi the piocess, there will bo a coirespondmg reduction in / (See I 104) 76 THERMODYNAMICS fen. Moie accmate values of cf) w aic obtained by nsin<> a foinmla devised by Callcndai which will be given when the deuvalion ol In*, tables is described (Chap VIII). In the tables Ihcic is n column foi the entiopy of watei at vauous tempcratmes, Ihc piessiiu in each case being the satuiation piessiuc at that lempcialme. It is the amount of entropy which the uaLei has at the end ol Ihc liisl stage in Art 56. when steam is just about Lo be 1'oimod During the second stage an additional amount of heal L is I;ik< n in at constant temperatuie T s , namely the Leni])cuiliiio .il uhich steam is formed under the given picssuie Hence ih< cnhopy increases by the amount ^- , and we have, foi Lhc cnhopy ol * 4 satiuated steam, Values of <^ s aie gi\ en in the tables Dining supei heating theie it> afiuLhei uicicase ol cnliopy .is I lie substance takeb in moie lieat. The cntiopy ol snpei healed slc.un at vauous piessuies and tcmpeiatuics Mill be found in OIK ol llu tables. It can be calculated by means of a foinuila \vlnch \\ill be given latei 74 Mixed Liquid and Vapour. Wet Steam In in my ol flic actions that occiu in steain-cnginct, and iclnyi inlinn machines uc have to do not with diy saLuiaLcd \apom bill \\ilh a imxhiic of satiuated vapoiu and liquid In the e\liiulci ol a sIcani-cMnmc, foi example, the steam is generally wet, it contains a piopoi lion ol watei which^anes as the stioke piocccds. Whuian\ such niivlim is in a state of theimal equihbiium the liquid and \apom li i\ c I IK sametempeiatuie, and the vapoiu is salinalcd. \Vlial is c,ill< d lh< diyness of wet steam is mcasmcd by Lhc fiacfion q ol vapoiu ulu< li is present m unit mass of the mixLiue When Ihc diyncss is known it is easy to deteimme othei quanhUcs Thus, Kckomng m c vci> case pei unit mass of the mixluic, we have Latent Heat of wet steam = qL = q (/., ~ J ttl ) ( [ ), Total Heat of wet steam, I q = I w + q L - / s - ( I < { ) L (a), Volume of wet steam, V, L = qV, + (1 - f/ ) V m (,{), \\hichibveryneailj equal to qV B unless Ihcmivlim is so \ u | as to consist mainly of watei , Entropy of wet steam, ^ = ^ + ( ^ ^ _ ^ ~ '/) L ( , } *- S -L j j. J-^V^JL J.JJH.J. j.J-ikJ VJL JL'JLiUlJJ'D 77 lorn (2) it follows that when the total heat I a of wet steam is lown, the dryness may be found by the equation T T r , _ ts. 1 (K\ 1- T _r ' ()- L a -* , >mbinmg (2) and (!), and eliminating <?, we have I, = / + T s (</, - w ) (6), Inch is a com ement expiession foi iindmg the total heat of wet jam when Lhc data aie the tempeiatine and the entiopy. An teinativc foim is jQ=J.-Z\(k-&) . . (7). these t:\picssions /, is the total heat of water, and /, that of y sal united steam, al the tempeiatine of the wet mixtiue. All these fonnulas apply to a mixtuie of aii3 r liqiud with its pom 75 Specification of the State of any Fluid We have now oken ol the follouin^ quanLiLics, \\hich aie functions of the state the substance Thcv all depend on the actual state, not on how dt sLilc h.is been leached The tempt Kitiue, T The pitssinc, P The volume, V The InU'uuil EIKIUN These foiu aie icckoned pei unit The ToUil IIe.il, / j quautitv ol the j, u butane c The Enliopv, f/> J A subslanee may change its state m main dillcient \\a\b it may inslanee take in heal al constant volume or ^hile expanding, ma\ expand 01 be computed Mith 01 without taking m heat, pansion ma\ lake place tluouyh a Ihiottle-vah e 01 nuclei a ,Lon BnL in any change of state whatcvei, the amount by which jh of these quantities is alteicd depends only on what the initial d final slates aie, and not at all on the paiticulai piocess by uch the change of stale has been effected Plicic aie olhei quantities, such as the heat taken in, 01 the work nc, which depend on how the change of state has taken place, dealing with them we have to distinguish between one piocess change and anolhei, even when both piocesses bung the sub- nice fiom the same initial to the same final condition. 78 THERMODYNAMICS n-n. The woikmg substance may be a liquid, 01 a mixluie of liquid and vapour, 01 a diy-satuialed 01 supeiheated vapoiu. The condition of a dry-satmated vapour is only a boimdaiy concliLion between that of wetness and that of supeiheat. To specify completely llu state at any instant it is enough to give eithei the piessuu 01 I lie tempeiatuie and one of the othci foui quantities named in Mils lisl. Thus if P and V are gnen the state is full}'- defined all I he ollici quantities can then be dctcimmed, pioMdcd, of comsi, wi- IUM- sufficient experimental knowledge of Lhe chaiaclcuslics ol 1 I he substance. Oi we may specify the state by giving auolhei pair ol quantities, such as T and <f> 01 P and /, 01 </> and /. Moie Generally, any two of these six quantities will sei\ r as d.tf.i in specifying the state, so long as the substance is homogeneous, but when the substance is a mixluie of liquid and vapoui [hi piessme and tempeiatuie do not suffice without some olhci pai- ticulai such as the diyness q, With legaid to these functions it may be useful Lo icpeaL lic-ic that T is constant in isotheimal expansion, </> is constant in adiabatic expansion: / is constant in expansion thiough a tluotllc-vah c or poious plug. 76. Isothermal Expansion of a Fluid Isotheimal Lines on the Pressure-Volume Diagram. A saturated vapoui can expand isotheimally only uhen it is wet the pioccs.s conesponds to I IK second stage in the expenmcnt of Ait. 56, it goes on nl conslanl piessme and invohes change of pait of the liquid in llu \vi I mixtme into vapoui Sinulaily, isotheimal com])ics,su,n ol .1 we I vapoui imolves condensation of pail of it Isolhcimal IIIH-S on llu- piesbure-volume chagiam foi a mixture of vapoiu and liquid ai straight lines of unifoim piessme. It is instinctive to considei the genoiul form ol (lie isollimn.il lines as the fluid passes successively thiouoh the sl.j-.i-s ol Ixn,.. (1) entnely liquid, (2) a mixtuie of vapoui and hquul^;}) ( ul nHv vapoious, by having its piessme giadually icducid iimiu- con- ditions such that the temperature lemains eonslanl lluounho,,[ | h, piocess Imagine for instance a cylmdei to conla.n a q,unl.ly of the liquid undei piessme applied by a loaded pjslon, mid le-l (he cylinder stand on a body at a definite constant Icnipcial.uo wluc-l, wU supply enough heat to it to maintain the lempei.-Uuu- un- changed when the pressme of the piston is giaduullv iclaxed and n PROPERTIES OF FLUIDS 79 the volume consequently increases. Starting from a condition of veiy high pressuie, say at A (fig. 14), when the contents of the cylinder are wholly liquid, let the load on the piston be slowly i educed so that the piessure gradually falls. The contents at first remain liquid, until the pressuie falls to the satuiation A^alne for the given tempeiature, namely the piessuie at which vapour begins to form. Thus we have in the pressme-volume diagram a line A 1 B 1 to icpiescnt what happens while the pressuie is falling during this fust stage, the contents are then still liquid. The volume of the liquid inci eases, but only ACiy slightly, in consequence of the piessuie being iclaxed, and hence A 1 B l in the diagram is ncaily but not quite vertical At B t vapoiu begins to foim, and continues Conning until all the liquid becomes vapour This is icpresented l>3 r B-iC-t, a stage clmmg- \diich there is no change of piessuie At (\ the ic is nothing but satuiatcd -vapoui Then, if the fall of piessuie continues, a line C 1 D i is ti.ieccl, the piogicssne lall >!' piessuie being associated with a piogiessivc incieasc of .olumc The tcmpeiatuie, by issiunption, is kept constant hiongliout. At I), , 01 at any )omt beyond C lt Lhc \apour MS become supciliealcd, bc- ause Us piessuie is loxvei than lie piessuie coiiesponding to .ahiralion, and hence Us lem- )ciatme it, highci than the empcialme coiicsponchng to almahon at Ihc actual pics- uie Anysuch line^/ZJC'Zhs an solhcimal foi the substance in he successive states of liquid A lo B), liquid and vapom nixed (B to C), satuiatcd 'apour (at C), supei heated vapoui (C to D) Now take a much nghci tempeiature We get a similai isotheimal A Z B 2 C 2 D 2 , and it a still highei lempeiatme anothei isotheimal A^B 3 C 3 D S , and o on. The highei the tempeiature the neaier do B and C ipproach each othci, and if the temperature be made high enough he horizontal portion of the isothermal line vanishes. VOLUME Fig 1 1 Isollioj nial Lines 77 The Critical Point: Critical Temperature and Critical Pressure. A curve (shown^by the biokcn line) duiwn thiou<>h 5^,63, etc. is continuous with one passing tlnough (\(\C^ and it is only within the region of which this cm vc is the nppei boundary that any change fiom liquid to vapoui taLes phcc. The bunch B 1 B 2 B 3 , which shows the volume of the liquid, meets the Inaneh C^Cj, uhich shows the volume of the saluraled vapour, in a jounded top. The summit of this cui\c icpicscnis a slate wlucli is called the CnticalPomt The tcmpeiatmc for an isolhcinial line E that would just touch the top of this cmve is called the Cnlicul Tempeiatuie We might define the ciitical Icmpeiatmc in miolhei way by saying that if the tempeiatuie of a vapour is above llu cntical tempeiatuie no piessme, howcvei gical, will cause it In liquefy. The piessuie at the critical pond is called the Ciitnul Piessuie, at any higher piessuie Lhc substance cannot exist as .1 non-homogeneous mixture, paitly liquid and paillv v.ipom. Staiting fiom D and mci easing the picssme, Lhc tenipeialuu being kept constant, ^e mav tiace anv of the isolhemuils luck- waids The initial state is then that of a gas (asupcihealed v.ipom) If the tempeiatuie is low enough \\c have a discontinuous pnx iss DCBA as the piessuie incieases C is i cached whin the \apoui is satuiated and condensation begins at B condcnsalion is eoiupkU , and fiom B upwards towaids A we aie compicssing hrjiiid A I any point between C and B the substance exists in two si, -i Irs ol aggiegation, pait is liquid and pait is vapoui But it UK- hm- peiatme is above the cntical tempeiatuie the isoMu-inul is OIK that lies altogether outside of the boundary curve, shown by llu- biokenlme, m that case the substance docs not siift'ei ,my sluup change of state as the piessuie uses. It passes fiom the sl.ilc' of a gas to that of a liquid in a continuous mannci, tollowmo a ( . ()IIIS( . such as is indicated by the lines F or G, and at no sl,,"^ m ,| u . piocess is it other than homogeneous The critical tempeiatuie foi steam is aboul .'](i;V C , ,md (he coiiespondmg piessuie is about 2950 pounds pu squa.e' inch In the action of an ordmaiy sbeam-engine the ciideal pou.l is m-v el- approached. But with caibonic acid, whose critical lunper.huv is only about 31 C , the bchavioui in the nciuhbo.nhood o! the cntical point, and above it, is of gieal practical nnpovtanee , connection with refngeiatmg machines which employ cai-bome acid as working substance. Gases such as air, hydiogen, oxygen and so foilh, are vapours PROPERTIES OF FLUIDS 81 which under ordinary conditions aie very highly supeiheated. Their culical temperatures are so low that it is only by extreme cooling that they can be bi ought into a condition which makes liquefaction possible The cutical temperature of Irydrogen is 211 C 01 32 absolute. Even helium, the mobt icfractory of the oases, has been liquefied, but onl}- by cooling it to a tempeiatuie within about 5 degrees of the absolute zeio 78. Adiabatic Expansion of a Fluid When a satin ated /apoui expands achabatically it becomes wet, and if it is initially ,vet (unless very wet*) it becomes wetter Its temperatme, Hcssurc, and total heat fall The fact that its entiopy remains maltcied allows the change of condition to be investigated as ollows, if we assume that the liquid and vapoui m the mixture ue in theimal equilibiuun throughout the piocess. For nieatci oenciuhtv we shall suppose the vapoui to be wet to >ct>m wilh Let the initial Icmpeiatine be T l and the initial diy- iess r/ L In Uus slale Ihe cuhopv is t i , <h L i 9 = ( Pwi + r r 1 - 1 1 JY b(ini> Ihe latent heat ol the \apom and <^ x the entiopy of the iqmel, bolh al Ihe lempeialme 7\ These quantities aie, tound in IK lables Lei llu siibslancc expand adiabaticallv to any loMer rnipcMlinc 7\, at \vlueh Ihe latent heat is L, and the cntropv oi he. li(|iiul is r/; H)i \u- ha\ c to find the usiilliiiy \ alin ol the di} ness, may n<>\\ be e\pn sscd ,is nd since Iheie IMS been no ch.'ini;i of entiopy this is equal to the nhal value c/; Lfence T > his e([U.ilion serves lo delei mine the chyncbs attci expansion, and nte il is known Lhe volume /'"" is icadily found as in Ait 71 Its v.u-t value is q z V^ H- (1 - q z ) F 10j , which is piactically equal in rtlmui y cases to g V^ , V^ being the volume of saturated vapour at * Whou tho nn^Luio is voty wcL to bogia with, ifc Tbocomes dnei during adiabatic panaion, beoauao so muoh of tho portion winch was initially liquid vaponzea under o reduced pressure that this moio than makes up for condensation in thepoition uoh was initially vapour (HCO Art 100). B.T. 6 82 THERMODYNAMICS [cu the temperature T 2 The pressure is the saturation pressure, corre- sponding to T 2 Thus the calculation fixes a point m the aeliabatic line of the pressure-volume diagram, for expansion from the iiuli.il conditions. A senes of points may be found in the same way, conei- spondmg to successive assumed temperatures which aie rcaclred in the course of the expansion, if it rs desuccl to trace the line. In the specral ca^e when the vapoui rs dry and saturatcel to begin with, the constant entropy (j> is equal to^., and the expression foi the wetness after expansion to any temper atuie T 2 becomes As an example of the calculation, let steam initially ehy and satuiated at a tempeiature of 190 C (P 1 = 182 1 pounds per sej "V.**- inch) expand adiabatrcally to a pressure of one atmosphcic ( Lem- per ature 100 C). The entropy, which remains constant duiui<> expansion, is 1 5013, </> Wi) is 3119, andI/ 2 rs 539-3. With Lhesc dal.i q 2 is 804, 136 per cent of Lhe steam has become liquefied, and Lht volume which was originally 2 534 cub ft. per Ib. rs 23 157 cub It. after expansion. Similarly, if the substance is entncl^ liquid in the initial stale-, the picssuic being sufficient to prevent vapour fiom foi mmi>, aeliabatic expansion will cause some of iL to vaporize Its initial entropy is iw , and since this does not change, J- > / i i - after expansion Lo a tempeiaturc T z Thus, vvjren watei initially at 190 C., and aL the coiicspondm<> saLination picssuic of 182 1 pounds pe j i sq. inch, expands adia- batically to a pressure of one atmosphere, q, becomes 0151 in othei woids 15 per cent, ol the watci vaporizes in consequence ol the expansion. The resulting volume is 4-127 cub. ft per Ib Conversely, il the wet mixture in this condition wcic compicssc d aehabatically it would become wcttci dunng compression, and | would be wholly condensed by compicssioir when Lire pressure reached 182 1 pounds per square inch. An approximation to the foim of the picssnrc-\olume curve for the achabatic expansion of wet steam is sometimes obi tuned by using an equation of the typcPF 1 " =constanL, and selecting a value of the index m appropriate to the initial state. Zcunoi gives foi 7- the index in Lhe formula m = 1 035 + Iq wheie q is the chyness [J PROPERTIES OF FLUIDS 83 t the beginning of expansion. But the use of such a method is un- atisfactoiy, foi a cm ve whose equation is PV m = constant, starting lorn any given initial condition, will agiee with the actual adiabatic mve at one other point only. The ciuves cross at that point. L value ot m can be selected which will make the equation >j7/n _ constant give the light volume when the piessuie has fallen om the initial pressure P l to any assigned piessure P 2 For this m pose we may wntc log PI - logP, logF a -lng?Y i which F 1 icpicsents the initial volume and F 2 the actual volume ached by expansion to P 2 F 2 being dctei mined by the method ready given, namely b> fii^t finding the dij-ness and then calcu- ting the volume fiom that It will be found that the \ alue of m rtained in this way becomes less the fuithei the expansion is nied, and also that it is oieatci when the steam is initially diy an when jt is Met To take an example, let the initinl state be at of diy satmatccl steam at 200 C , loi which the picssine is 521f pounds pei square inch Adiabatic expansion to \aiious wei Lempci.itines and piesbiiies gi\cs the tollowmg lesults Tornpoi iliac Piossuio Volume Index. C pounds pci MI in Divac-js (nib feet) m iLial) 200 22524 1 2 073S 190 18208 9828 2 4906 1 162 170 1150G 9525 3717 1 151 140 52 48 9085 7 400 1 145 100 14 69 8528 22 849 1 138 40 1 07 7693 240 35 1 126 each stage the calculated \ aluc of m is that which would make a \o having the equation PV m =P ] F 1 " i pass through the point ched at that stage m adiabalic expansion The Zeunci foimula, >tcd above, makes the index loi initially diy sic am 1 135, but t would make the CUM e he too high in the cail\ stages and too r alter the piessuie has fallen below one atmosphcie No constant ex can give a leally good appioximation to the actual cm ve. !o fill, this Article has dealt with the expansion of satmated or , vapom. When a supei heated vapour expands achabatically its ansion is divisible into two distinct stages The fust stage bungs own to the state of satmation, m the second stage it is a Met torn and the foregoing methods of calculation apply. Callendar 62 84 THERMODYNAMICS [en. has shown, that, m the firsL stage, supei healed steam expanding adiabatically follows closely the equation p(V - 6) 13 = constant, wheie b is the volume of watei at C , namely 010 cubic i'L. per Ib Except at high pressuies b is negligibly small compaied with V, and may be omitted without serious erior. This equation applies down to the point at which the s Learn becomes saturated. The amount of adiabatic expansion which will bung a supci- heated vapour down Lo that point is detci mined fiom the fact that the entropy is constant. We have only to find at what Lcmpciatmc (or piessure) the cntiopy of saturated vapoui is equal Lo that of I he supei heated vapoui in the given initial state. This companion is icadily made when tables 01 charts are available giving the pio- perLies o the substance in both states, supeihcaled and satin. iLcd. Callendai's tables give the necessaiy data foi steam then use in such calculations will be illustiatcd in the nevt chaptci The ch.uls which will ako be desciibcd there seive wcllfoi the examination ot cases in which the vapoui is supei heated bcfoie expansion. Such casei. occui fiequently in steam-engine pi ac I ice With a suitable chait it is easy to tiacc the whole coin^e of any adinbaLic expansion tlnough the legion oi biipeiheat. past the point of satin alum, and finally in the legion of wetness It must not be supposed that the expansion of blcnm in an ad ual engine is adiabatic, foi there is always some Iransfci of hc'.il be- tween the working fluid and the metal ol the cyiindei and pislon If it weie practicable to use a peifectly non-conducting male ual foi the suifaces m contact with the steam, the ideal of adia- batic expansion could be icalizcd. It is appicmmaLcd lo in cases wheie the action occins too fast to allow any considerable h.mslVi of heat to take place Sudden, and theielore practically aduibatic, expansion luun a high piessuic may be used to produce a veiy low tcmpcialmr ll was in this way Lhat ga,set, such as oxygen and nitiogen wcic loi I lit, first time liquefied The gas was compicssccl and was cooled in I he compiessed state to a fanl3 r low tcmpciatuie. It was then suddenly expanded, and the further cooling which resulted fiom I his ex- pansion caused a poition of it to become liquid. 79. Supersaturation In discussing adiabatic expansion we have assumed that theie is at evo^ step in the expansion a con- dition of equilibrium in the fluid, that is to say equilibrium between [] PROPERTIES OF FLUIDS 85 tie part that is vapour and the part that is liquid. But it is known, s a icsult of experiment, that when a vapour is suddenly cooled y adiabatic expansion the condition of equilibrium is not leached t once. Suppose the vapour to be initially dry and saturated : on xpansion a pait of it must condense if eqmlibimin is to be eslab- shed. This condensation takes an appieciable time, it is a suiface henomenon, taking" place partly on the inner surfaces of the Dntainmg vessel and paitly by the growth of diops thioughout ic volume. Consequently the sudden expansion of a vapour may roduce tempoiaiil}'- a condition that is called supei satui ation, i which the substance continues foi a time to exist as a homo :neous vapour, although its pressure and tempeiatuie aie such lat the condition of equilibrium would lequne a pail of it to be nidenscd In this supersatuiated state the density of the vapour abnoimally high, higher than the density of satuiated vapoui at ic actual picsbiue The temperature is also abnoimally low, lower tan the tempeiatuie of satuiation at the actual pressuie for this ason the supeisatuiated ^ apom might be called supercooled The ipersatmated condition is not stable it disappeais through the mdensation of a pait of the ^ apom, and the lesulting mixtuie of ipoui and liquid has its tempciature laiscd In the latent heat Inch is given out in this condensation We shall sec latei, in nncction with the theory of steam jeti. (Ait. 135), that cx- msion involving supcisatmation may occui under piactical ndihons*. The supeicoohng of a \ apom without condensation is analogous the supei cooling of a liquid without crystallization In both scs thcic is a clcpailiuc horn I he slate of cqmlibinim, and in both scs the icstoialion ol cqiiilibiium invokes an nieva stble action, thin I he substance The noimal adiabatic expansion of a \ apoiu, alt with in Ait 78, is icvcisiblc, but if theie has been super- olmg theie is an iue\ eisiblc development of heat within the fluid ' An mtoiostmg oxnm[>lo of supeisaturation occms when dust-fieo air saturated Ii water vapom is suddenly expanded So long as paiticles of dust are piesent nsfc foims (on slight expansion) by the condensation of water on them as nuclei, r if they ato lomovod bofoio such an expansion the nifst does not form and the >our becomes supoisatmatod If however the ratio of expansion is laigo, so that re is much supeicoohng, a mist forms even in the absence of dust in that case ppoais that diops of the liquid form about smaller nuclei; which ate not of foicign ttor, but prob.ibly consist of gioups of molecules accidentally linked in the course ho moloculai collisions that occui m any gas (See Aitkcn, Tians RS E vol 30, Nairn a, March 1, 1888 and Fob 21, 1890 also C. T R Wilson, Phil Tiam S. vol 189, 1897 ) SO THERMODYNAMICS [en. when the supercooled vapour passes into the stable state of a mixture of liquid and saturated vapoui. Duung supeicoolmg by adiabatic expansion steam expands according to the formula P (V b} 13 = constant. The foimula is the same as for the adiabatic expansion of supei heated steam. IL applies whethei the steam be initially superheated or saturated, and continues to apply so king as the steam- expands in a homo- geneous dry state, as a icsult of supei saturation. 80. Change of Internal Energy and of Total Heat in Adiabatic Expansion "Heat-Drop." When a fluid expands acliabahcally from any condition a to any othei condition b Hie decieasc of internal eneig}" E a E b is equal to Ihe thcimal cqmva- lent of the woik done in the expansion This is because it takes in no heat and consequently the work which it docs in expand- ing is done at the expense of its stock of internal cneigy. Refeiung to the piessurc- volume diagram (fig. 15) the woik done dining expansion fiom a to b is measuicd by the area inabn, consequently in _ adiabatic expansion Volume T-I n , , 7V T^Jff JO E a Ej, = A (aiea wnbu} Further, the decieasc of Lolal heat which the substance umlu- gocs din ing the pioccss is equal to the Ihcnnal cquiMilcnl of I lie aiea cabf To pio\e llus, \vc have, by the definition of the tol.il hqat(Ait. 07), r _ J7 I ///> y 1 ti ~ lj a r -" ' j and 7 ft =/(l 6 + JZ' t r 6 , fiom Avhich m = A (aiea mnbn -i- aiea camo a = A (aiea cabf) This is tnic whatever be the condition of the fluid befoi c expansion it applies for example to superheated as well as lo satin a bed or wet steam, 01 to any gas PROPERTIES OF FLUIDS 87 It may be instructive to the sbudent to have the same pi oof put in a somewhat different foim. From bhe equation winch defines the total heat I in any state, namely, I = E + APV, \ve have by differentiation dl = dE + Ad (PV) = dE + APdV + AVdP. Bub in any small change of state it follows from the conseivafcion of energy that the mcicase of internal energy plus the woik done by the fluid is equal to the heat taken in, or dE + APdV = dQ, vvhcie dQ is the heat taken m during the change. Hence in any small change of state rfj = ^ + AVdP. In an adiabatic opeiation c/<2 = 0, and hence in that case HI = AVdP. Therefore if the fluid expands adiabatically from state a to state b the resulting dccicase in its total heal, namely /-/ = A VdP. This integial is the aica cabf of the picssuie-volumc diagiam 'fig 15) It is the \\hole \\o\k done m a cslmdcr when the lluid is admitted at the picssuic corresponding lo state a, then expanded ichabatically to slate b, and Hun disehaiged at the pitssiuc conc- ipouding lo slale b The decrease of total heat m expansion, I a I b , is called Ihe "Ilcat-dio]) " It is a quantity of much importance in the lluoiy of heal -engines The aboye equation sho\\s that under idiabatic conditions the whole work done in the cyhndci, when expressed in heat units, is measured by the heat-drop In the next chapter this principle will be applied to infer from the heat- drop the work that can be done in steam-engines under \anous assumed conditions, and it will be shown how to calculate the heat- drop which occurs in adiabatic expansion from anj initial state CHAPTER III THEORY OF THE STEAM-ENGINK 8 1 Carnot's Cycle with Steam or other Vapour foi Work- ing Substance. We aie now in a position lo study (lie .id <>l a heat-engme employing watci and steam, 01 any olliei li([iii(l .md its vapoui, as the ^oiking substance To siiiiplil'v (lie lusl coii- bidciation of tlie subject as fai as possibk, hi il be suppose d (ha I we ha\ e, as befou, a long o^Inuki, composed ol nou-conduel m^ matenal except at the base, and filled \\ilh a ii<>n->iidiiel inn piston. albO a souice of licat A at some lunpt i.il me 7 1 , , .1 K eei\ 1 1 of heat, or as we may now call it, a comic usci, (', al some lo\u i tempeiatuie T 2 , and also a non-conduclmi> cm ei II (.is in Ail .><>) Then Cainot's cjcle oi opeiations can bo pulonmd .is l<>||o\\s To fi\ the ideas, suppose that then, is uml m.iss <>l w.ilu in llu cyhndci to begin with, at the tempeuilim 7', (1) Apply A, and allo^ the piston lo use ai>amsl (In c-onsLml pleasure P l which is the satmalion piessmc coiiesponduin | ( , UK tempeiatuie T T . The water will take m he. it .md IK eonxdUd ml<. steam, expanding isotheimalh .il tlu lempc lalme 7', Tliisjt.iil ol the opeiation is sho\Mi by the line ab in hi- 10 (2) Remote A and appl\ B Allow the cvp.msiou |> , |,,,i, adiabatically (to), with falling pics^me, unlil Ihe U mpi i. ,(,i l( |.,|| s to T 2 The piessmc will then be P 2 , nmncly, llu piessnn \\hi< h coiiesponds in the steam [able to 7\, uhich is (lit l(miKi..lun ,,l the cold bod\ C (3) Re-motel?, apply C, and compiess Sle.im is eondens. d l,\ lejectmg heat to C The action is isolhcuiu.1, and ||, ,,,,SSI.M lemams P a Let tins be contmued unhl ., c-c,l,m, p.,,,,1 ,/ ls leached, hich is to be chosen so th.,1 ad,ab,,lie eonmu ss,<,n ,|| complete the cycle (4) Remove C and apply B Continue the compiess,,,,, win. I, is no adiabatic If the point d has been noluly c-hoscn, || lls n,|| complete the cjcle by ^stoung the woikmg || llld , () lhc sla|r <)( Mater at temperatuie T T . r. in] THEORY OF THE STEAM-ENGINE 89 The mdicatoi diagiam for the cycle is drawn in jfig. ifi, the lines and da having been calculated by the method of Ai t. 78, for a irticular example in which the initial piessnre is 90 pounds per nave inch (2\ = 433), and the expansion is continued down to the essuic of the almospheie, 14 7 pounds pei squaie inch (T 2 = 373). Fig Id Gi mol's r\clo \\illi Nfitei and steam lui \voikiny mibslanco iincc I lie pmcess is K \usible, and since heat is taken in only at and ic|cclcd only at 7\, Ihc efficiency (by Ait. 38) is c lie.it taken in pei unit mass of the liquid is L 1} and theici'oic woik clone is T ^sult which mny be used to check the calculation of the lines Lhe diagram by compaimg it with the aica, which they enclose .vill be seen that the whole opciation is stnctly icveisible in the rmodynamic sense. 90 THERMODYNAMICS [cu. Instead of supposing the working substance to consist wholly of water at a and Avholby of steam at b, the operation fib might be taken to repiesent the paitial cvapoiation of what was originally a mixture of steam and watei The heat taken in would then be (?& ~ <Ja) ^1 an d as the cycle would still be reveisible the aica of the diagiam would be 82 Efficiency of a Perfect Steam-Engine. Limits of Tem- perature If the action heie descubcd could be reali/cd ID piactice, we should have a thcrmodynamically pcifcct slcjun- engine using sarmatcd bteam. Like any other pcifect heat-engine, an ideal engine of this kind has an cmcicnc} r which depends upon the temperatures between which it works, and upon nothing else The fraction of the heat supplied to it which such an engine uould convert into woik would depend simply on the two Icmpcialmrs, and Iherefoie on the picssurcs, at which the steam A\as pioduoecl and condensed icspectively. It is inteie.stmg theicfoie to considci what me lhc hmils of tempeiatuie between which s team-engines may be made to \\oik The tempeiatuie of condensation is limited by Ihe considcinhon that theie must be an abundant supply of some substance lo absoib the lejected heat, watei is actually used foi this puiposc, so that T 2 has fonts lowei limit the tempeiatuie of the available watei -supply To lhc highci temperature 2\ and picssuie />, a piadical linn I is set by the mechanical difficulties, with icgaid lo sticnglh and to lubrication, which attend the use of high-picssmc steam In steam motor-cais piChSiues of 1000 pounds pci sq inch have been used, but with engines and boileis of the ouhnaiy consliuchon the prcssmc langcs fiom about 300 pounds pei sq inch clowns ,uds This means that the upper limit of tcmpcrahue, so fai as satin - atcd steam is conccincd, is about 215 C A slcam-cnginc, Iheie- forc, undei the most favouiablc conditions, comes very I'm shoit of taking full advantage of the high tcmpciatinc at which lu<U is pioduced in Ihe combustion of coal Fiom lhc thcimodynanuc point of view the woist thing about a slcam-cngme is the me- versiblc diop of tempeiatuie between the combustion-chambei of the furnace and the boiler The combustion of the fuel supplies heat at a high tcmperatuie but a gieat part of the convertibility in I THEORY OF THE STEAM-ENGINE 91 of that heat into woik is a I once sacrificed by Ihc fall in temperature which is allowed to take place before the conveision into work begins. If the tempcratme of condensation be taken as 20 C., as a lower hmil, the cflieicncy of a pcifcct s team-engine, using satmatcd slcain and following the CainoL cycle, would depend on the value of I\, the absolute picssmc of production of the steam, as follows: Pcifect steam-engine, with condensation a,L 20 C., P t in pounds per sq inch being 50 100 150 200 250 300 Highest ideal eflicicncy - 2SS -330 355 373 384 399 These numbers cxpicss what fiaclion of the heat taken in by the woikmg substance would be comcitible into woik under the ideally favouiablc conditions of the Cainot c^ clc But it must not be supposed that these \dlues of the eflicicnc}'' .no actually attained, 01 aic even attainable Many causes con- spue lo pie\cnt sic.un-engines fioni being tlicimodynainically |)ci led, and some of the causes of nnpti (ctlion cannol be iemo\ cd Tlu-sc nmnbeis will seivt, howe\ci, as one srand.ml of companion in judging of I lie pc i Joimancc of ae( n.il engines, and as illiistiating the .uhanlage of Iiigli-j)iessuie slcam fioni Hie lliennod} nanuc ])omlof\nw. We shall see m Ai I 87 that llicie is anolhu .slanchud wilh winch Ihc pcil'oinianee ol a ical sicam-cnginc may moie appropiulely be complied 83 Entropy-Temperature Diagiam for a Perfect Steam- Engine The nnagmaiv sk.mi-engmc of Ait. SI lias the same vei y simpk enliopy-lenipcialiiic di.igiam as anv olhci engine which follows C'ainol's C) cle Tlie fom opcialions aic icpiescnted by the fom sides of a leclanglc (fig 17) The fust opeiation ch,mg( s watci (at the upper limit of tcmpcratmc) into saturated slcnm at Ihc same tcmpciahu-e, the enhopy accoidingly changes Iiom (/}, to <:/>,, This is shown by the conslmit-lempcia,tuic line fib in Jig 17 In the second opeiationAvhich is adiabatic cx- pansion the cnlio})y docs not change, and the tempcialurc falls lo the lowci limit, at which heat is to be icjccted this is repic- sented by the line of constant cntiopy be In the thud operation, cd, the Icmpeiatme jcmams constant and the cntiopy is icstoied lo its oiigmal value, heat being rejected to the cold body. In the fourth operation which is adiabatic compiession the entropy does not change, and the temperature rises to the upper limit. 92 THERMODYNAMICS [en. the substance has returned to its initial state in all lespects. In order to be compaiable with other diagiams which will folIoAV, fig 17 is sketched for a pal ticulai example in which P : is ISO pounds per sq. inch, and P 2 is 1 pound per sq. inch: consequent!}' Z : is 189 5 C and t z is 38 7 C g Expressed in tei ms of entropy, "J5 the heat taken m (during ab) is ^ TI (4> s <f>w)- Thisisicpiescnted by the aiea undei ab measured down to the absolute zeio of temperatme, namely the area inabn. The heat i ejected (dm ing cd) is T 2 (^ (j> w ) and is icpie- scntcd by the aiea ncdm. The ff ' Entropy n theimal equivalent of the woik Fig. 17 done in the cycle is accoidmgly (T l To) (</> s cf> lu ), and is icpiescnted by the area abed, enclosed by the lines which icpiesont the fom icvcisible opciations. The efficiency is ^ _ T ,) (fa - <ftj _ I\ - T. 2 In the example foi which the diagiam is diawn, with Ihe dala stated above, the numciical value of this is 326. 84. Use of " Boundary Curves" m the Entropy-Tempera- ture Diagram. In fig IS the diugicim of /ig 17 is diawn ovc-i again, with the addition of a cuivc Ihiough a which icprcsenls Ihe values nl vanoiis tcmperatuies of </>,, the cntiopy of watei when steam is just about to fonn, and a cuivc Ihiough b which icpie- sents at various lempciatuics the value of </>,, the entiopy of cliy satiuatcd steam These curves aie called Bouuddiy Cmvcs They aie icadily drawn fiom the dala in the steam tables Any point on Ihc boundary curve thiough a would relate to the cntiopy of water, between the two curves any point in Ihc diagiam relates to a mixtiuc of watei and steam, to the right of the boundary curve Ihiough. b any point would i elate to steam in Ihc supcihcatcd state. We aie not at picscnt concuncd willi the outlying regions but only with the space between the two curves, willnn which the points c and d fall Let the line cd be pioduccd both waj T s to meet the Ill] THEORY OF THE STEAM-ENGINE 93 I- boundary curves in e and s. Then the ratio of cs to es represents the fraction of the steam which becomes condensed during the achabalic expansion be from the condition of saturation at b To prove this we may first consider the meaning of any hori- zontal (isollicimal) line such as se on the cntiopy-tempeiatnie diagram between the two bomiddiy curves It ic- picsents complete con- densation of 1 Ib of dry satuiatcd steam, under constant tempeialiuc andpicssurc Dining Us convcision fiom the con- dition of diy saturated steam (al &) to water (at e] the steam gives out a quantity of heat which is measured by the aica undci I he line, namely the aica of>cl Any inter- mediate point in Ihc line icpiLscnls a mi\line of water and steam, llms c icpiescnts a mixtmc which, though it has actually been pioduecd by adiabatic expansion fiom b, might have been pioduced by puitial condensation fiom 6 undei constant pressine, apioecss which would be u'prchcnled by AC, 01 by partial cvapoiatiuu under the same constant picssuie fiom e, a pioeess which would be icpicsentcd by cc. Now if the nnxluie at c, weie completely condensed undci couslant pressuie to c, the heal given out would be mcasmcd by Ihc aica tied This heat is gi\cn out by the condensation of that part of the mixture which consisted of steam Hence the fraction which existed at c as steam, or in other woids the diyness of the nuxtuic at c, is measured by the latio of the aicas ncel to oscl, which is equal to the ratio of the lengths cc to cs Hence also the ratio cs to as mcasiucs the wclness of the mixture at c. An en tropy- tempera tuie diagram on winch the boundaiy curves aic diawn theieforc gives a convenient means of determining the wetness of steam at any stage m the process of adiabatic expansion. It is only necessary to chaw a vertical line through the point repre- senting the initial condition. That line represents the adiabatic Entropy Flit IS 94 THERMODYNAMICS [CH. process, and the segments into which it dnides a horizontal line drawn from one boundary curve to the other at any level of temperature represent the proportions of water and steam in the resulting mixture. Tins is true not only of the final stage, when adiabatic expansion is complete, but of any intermediate stage, for the argument gi\en aboA e obviously applies to a horizontal line diawn at any temperature between the two boundary curves. Srmilarly the point d which represents the wet mrxtinc at the beginning of adiabatic compressron da, shows by the ratro of segments ds to de what is the proportion of Avater to steam at Avhich the third stage of the cycle has to be arrested, in older that adrabatrc compressron may brrng the mixture Avholl}' to the state of Avater Avhcn the cycle rs completed by the operation da The student should compare this graphic method of studyrng the A\ r etncss resulting from adiabatic expansion Avith the calcula- tions given in Art 78 He Avrll observe that both have the same basis. At any temperature T the length es of the isothermal line draAvn from the water boundary curve at e to the steam boundary curve at s rs L/T, and the intercept cc up to any intermediate point c on that line is qL/T, Avhere q is tire di} ness of the mixtmc at the point c. The same principle ol course holds for the entropy-tem- perature diagiam of any other fluid. 85 Modified Cycle omitting Adiabatic Compression. Con- sider next a modification of the Carnol c\clc of Ail 81 Let the fust and second opeiations occur as thc} r do there, but let the third operation be continued until the steam is wholly condensed. The substance then consists of Avatcr at T 2 , and the cycle . _" * is completed by healing it, in the condition of \\alci, fiom jf' 2 to r l\ In the simple engine of Ait 81, \\hcre all the operations occur in a single k _ e \ cssel, this could be done by increasrng I he pressure exert- j, J0 . 1{) cd by the piston from P 2 to P 15 aftci condensation is complete, then icmoA'ing the cold bod} r C and applying the hot body A. The Avater is therefore heated at P! and no steam is formed till the tcmpcratmc reaches T 1 The pressure-volume diagram (or indicator diagiam) of a cycle Ill] THEORY OF THE STEAM-ENGINE 95 modified in this mannei is shown by abce m fig. 19 The sketch is not chawn to scale As before, ab is the operation of foiming steam, from water, at r l\ and P 15 be is achabatic expansion from T 1 and 1\ to T z and P 2 Then ce is complete condensation at T 2 and Po The fouith operation ea now involves two stages, fiist laismg the pressme of the condensed water from P 2 to I\ and then heating it fiom To to T t . Dining both of these stages the changes of volume aie negligible in comparison with those that take place in the othci opciations. The cntropy-tempciature diagi am ioi this modified cycle is shown by abcc m iig 20, whuc the same Icttcis as in fig 19 are used foi coiicsponding opciations As m the Cainot cycle, ab icpicscnls the conver- sion of a pound of watci at r l\ mlo diy saluialcd steam at 2\, and be ic- piesenls Us adiabalic ex- pansion lo 7\, lesnlling m a Met mivlnic at c, Ihc diyness of \\hieh is mpa- suied by Ihe latio tic/cs Then cc upiescnls the complele condcnsalion at T 2 ol Ihe sleam in llus wet inixlnu, and ca, Avhich ])iaclic.illy coincides with the boundaiy ( in \ c, icpie- suils llu le-luahng ol Ihe c condensed \val< i liom T> to 7'j , al'lci Us picssmc has been uuscd lo 1\ so Ihal no sleam is lormccl during Ihis opeialion ' The woiking subslnncc behaves rcvcisibly Ihroughout all these opciations, and Iheiefoie Ihe woik done in the cycle is repicscntcd by Ihe area nine in Ihe cntiopy-tcmpciatuic diagram of fig 20. The diagram further exhibits the heat lakcn m and the heat re- jected The whole heat lakcn m is mcasmed by the area leabn, and of this Ihe area ham mcasmes the heat taken m during the last * Tho lino ca in Loth dmgiams, figs 19 and 20, ically stands foi a broken ]mo la'a, whoio ca' loptowonts tho rawing of pressure fi om P^ to P : at constant tompera- buic TI, and a'a iopicnonts tho heating from T z to T 1 at constant pressure P lt In fig 10 a' pmctically comoidos with a, m fig 20 a' practically coincides with c m Entropy n 96 THERMODYNAMICS [CH. operation, while the water is being re-heated, and the area mabn measures the heat taken dmmg the fhst opeiation, while the waber is tuining into steam. The aiea ncel measures the heat re- jected, namely dining the condensing piocess ce. To express algebiaically the woik done in the cycle, icfer to the indicator dmgiam, fig 19, and let the lines ba and cc be produced to meet the line of no volume m j and k. Then, by Art 80, the aveajbcL is an amount of Avoik equivalent to the difi'cicncc of total heats 7. _ / -*& * CJ namely the "heat-drop" of a pound of steam in expanding adia- batically fiom the condition at b to the condition at c The small area jaeL is (P t P>) F Wa where V Wn is the volume of a pound of watei at T z , which we ma}'- take to be piactically constant ibi the piuposes of this calculation Hence the expression 7 6 - I e - A (P, - P 2 ) F B> is the thermal equivalent of the woik done in the cycle If figs 19 and 20 weie both caiefully diawn to scnle foi any paiticular example, a measiucmcnt of the enclosed aica abcc in cilhci figuic would give a lesult m agioemcnt wiLh this calciilalion. 86. Engine with Separate Organs The impoi lance of llie modified cycle descnbcd in Ait 85 lies in Ihc fact of Us being UK- Pjg 21 nearest appioach to the Carnot cycle that can be aimed at when the operations of boiling, expanding and condensing are conducted m separate vessels. The imaginary engine of fig. 16 had one organ mj THEORY OF THE STEAM-ENGINE 97 only a cylinder which also served as boiler and as condenser. We come neaier to the conditions that hold in piaclice if we think of an engine with sepaiate organs, shown diagiammatically in fig 21, namely a boiler A kept at T 1} a non-conducting cylinder and piston B, and a sniface condenser C kept at T 2 To these must be added a feed- pump D which ictiuns the condensed watei to I he boilei Piovision is made by which the cylinder can be put inlo connection wilh Lhc boilei 01 condenser at will With llns engine the cycle of lig in can be pei formed An in- dicator eliagiam foi the cylinder 7? is sketched m fig 22 Steam is admitted fiom Ihc boiki, giving Ihc line ]b At b "cut-off" occurs, lh.il is lo say the vahe winch admits slcam fiom the boiler to the cylnulei is closed The slcam in the cyhnclei is then expanded aduibalicdlly to Ihc piessiirc of the condenser, gu mg the line be Al ( the "exhaust" \alvc is opened vihich connects the cylinder with (he condcnsci The pislon then letmns, discharging the slc.un lo Ihc condense! and giving the line c/i The area jbcl l<V r 22 Fig 22 a epuscnls (lie 1 \\oik (loin- in llic eylindu li Tin. condensed wafci s Hun icliiuud In Hit 1 boilci bv llu fccd-pmu]), .nul Ihc mehcaloi lugiam showing Ihc \\oiL ixpcndtd upon Ihc pump dining Llns >pu,ilion is sUilcliul in dg 22 a II is Ihc icclangle hi'tt), when c icpicstnls Ihc np-shoki m which Ihc pump (ills with walei I the pitssmc /\, and <i) icpicscnls Hie dowu-sliokc in which it ischaigts walci lo Ihe boiler agamsl the picssurc Pj If we upupose Ihr (lugiuin oi Ihc pump on tluil of the cyhnelci we get IK u (lif('cuMic( , iiani( ly alitr (fig 10), lo icpiescnl the net .unoiint { work done by Lhc Hind m the c> clc. It is the excess of the work one by the /Imd m Ihe cylinder over that spent upon it m the nmp. Taking- the two parts sepa lately, the adiabatic heat-drop, * ~ -LI 98 THERMODYNAMICS !<'" is the thermal equivalent of Lhc woik done by UK- ilmcl in UK- cylmdei, and ^ _ p,) J/ UB is the woik spent upon the fluid in the feed-pump Acroidmidy the diffeience, namely I,-I c -A(l\-PJV Wi , is, as befoie, the theimal equivalent oi 1 (he woik obl.umd in Mie cycle as a whole. 87. The Rankine Cycle This cycle is commonly called I lie Rankine Cycle Like the Cainot cycle il i-cpu-suils .in uh.il Ih.il is not practically attainable, foi it postulates a complete absence of any loss thiough tiansfci of heat between llic sU.im .nul I he smfaces of the cylinder and piston. But it allouls .1 vriy v.iln.ible cntenon of peiformance by famishing a stand, nd \villi \\lueh UK efficiency of any leal engine may be compaiod, a slandaid \\lnch is less exacting than the cycle of Cainol, but 1. 11101 I'm companson, inasmuch as the fouith stage of Ihc Cunol cvcli 1 is luetss.mlv omitted when the steam is icmovcd fiom I he cylnuhi Ixlon con- densation. A sepaiate condensci is indispensable in an\ u al engine that pietends to efficiency The use of a sepaiate condensci was in lad one <>l UK HKM! mipio\ements \vhich distinguished the sle.un-en^iiK ol W.ill liom the eaiher engine of Newcomcn, A\heic I ho sli'.im \\.is (ondt nsi d in the working cylmdei itself The mliodnelion ol'.-i s<'p;n,il con- denser enabled the cylmdei to be kc-pl comp.iiali\ t-l\ lu>l, and theieby i educed immensely the loss lhal h.id o((iiii(d in <,nliu engines thiough the action of chilled cylmdei sml.icis upon (In enteimg steam. But a sepaiate condonsei, gK.illy Ilion^li il .idds to efficiency in piactice, excludes Ihc compu-ssion sl.in* <>l lh< Cainot cycle, and consequenlly makes Ilic Hankiiu c\ cli Ihc piopei theoietical ideal with which Uicjicifoinicincc ol'.i n-.il ( rmuu should be compaied. The efficiency of the Rankine cycle is less llun lh.il ol a Carnot cycle with the same Imnls ol Icmpi-ialnic Tins is because, in the Rankine cycle, the heal is nol all l,ik<'n in a( Hit- top of the lange In the Rankine cycle, as m Cm mil's, all Ihc intemal actions of the woikmg snbslame ,ue, by assiimplion, reveisible, and consequently each elejnenl of Ihc whole hcal- supply produces the gieatest possible mech.imcal elTrc'l when legaid is had to the temperatme at which lhal elemenl is Inkcn in. - ml THEORY OF THE STEAM-ENGINE \ ..\ J \ -r, \, V Ts' But part of the heat is taken in at temperatures lowcKtan %\> namely while the woikmg substance is having its tcmpGrattfre; f raised from T z to 2\ in the fouith opciafcion. Hence the average'- efficiency is lower than if all had been taken in at T 1} as it would be in the cycle of Carnot. "* pound of steam does a laiger amount of woik in the cycle than it docs in the CainoL cycle. Tins will be t when the uieas aic compaied which lepiescnL the work in Llic corresponding diagiams- the aica nbcc with the aica abed in lio 20. 13 ut the quantity of heat that has to be supplied for each poiuid in the Rankine cycle is also gicatci, and in a gieatei latio iL is mcasmcd by the aica leabii, as against mabn Ilcncc the efficiency is less in the Rankine cycle One may put Ihe same thing in a different way by saying thai, in the Rankine cycle, of the whole heat-supply the paitZra/M does only the compaiatively small amount of woiktw/, and the lemamdci oC the heat-supply, namely ma\)n , does the same amount of woikas it would do m aCainoLc>cle 88. Efficiency of a Rankine Cycle. Tjkmg in the fiist instance a R.nikine cycle in winch the steam supplied to the cUmdci is diy and saturated, tlic \\liolc amount ol heal taken in is the quantity loquivcd to convcit water ,il Pj and T 2 into satuiatcd steam at J\ Tliis quantity is / - {/ 1t , a + A (l\ - P z ) /' u , a }, Jbi the total heat oi Iho watci at P l and To is gieatei than l Ui by the quantity -/(/*! -^)^ a Tlic* woi k done is (by Ait. So) CCMI.I! lo tlic lu.ii-diop muni* the woik spc-nf- in the k-td-pmnp, 01 / - / c - A (I\ ~ 1\) l\ tt whcic / is I he Lolal licat of L ho we I mixluie aflu adiabalif expansion The* c'/Iic'iLiic'V in I lie c\clc as a \\hole is Lheicfoic The ieecl-pump U-im A (P l - P z ) V w , is iclalivcly so small that it is of Lei i omiUed in calcuJadons iclatmg to ideal efficiency., jiiht is it is omilU-d in staling the icsults of tests of the pcifoimance of L-eal ongiiies. In such tests it is customaiv lo speak of the work lone per Ib. of steam, without making any deduction for Ihe woik hat Jms to be spent per Ib in returning the feed-water to the loilcr. 13 ut in the complete analysis of a Rankine cycle the feed- >uinp term has to be taken into account, and il is only then that the i,rca of Hie entropy-temperature diagram gives a true measure of he work done. It should be clearly undei stood that the heab-drop, 100 THERMODYNAMICS [i it. by itself, is not an accurate measuie of the woik done in I he Rankine c} 7 cleas a whole, 1101 is the hcat-diop equal lo the ciu-Iosi d areaof the entropy- Lcniperatiuc dia^iam, until the thermal equn a lent of the work spent in tlioieo'd-pump has been deducted fioin il If however we aic concerned only wilh I he \\ork done in the cylinder of the ideal engine, then the heal -drop alone has ID U- leckoned. It is the exact mcasiue of lhal woik The ralio ol (lit heat-drop to the heat supplied shows whal piopoihon of I he supply is conveited into woik in I lie cylinder, under the ide.il conditions of adiabatic action it is a lalio nearly identical \Mlli (In ellieu nrv of the Rankmc cycle, and cvcumoio useful asaslandaid \\ilh \\lueh to compare the pciloimancc of a leal engine In (he .iclual pt i- formance of any mil engine Iho amounl ol uoik done in the cylmdci necc^anly falls shoj I ol Ihe adiabahc heal-diop IK can-. the woikmg substance loses some heal lo (he cylinder \\alls. TJu extent Lo which it falls shoiL is .1 mallei I'oi (ml, and IUM-( Ih.tl has been asceitamcd by luals of engines ol <.|\rn lyp< s, eshm.ih s may be made of the pcifoinuncc of an ennine IMH|< i d< SKMI nsni the adiabatic hcat-di op as Llie basis of (he c.ileiilalion, \\\\\\ \ -.mi able allowance foi probable wasle 89. Calculation of the Heat-Drop Ft is fheiefou ess.niml to be able to calculate the heal-diop in id<al CMIIUS und. i ., M \ assigned initial and hnal con- ditions. Foi thispiuposenc / \ have to find / c , the tolal c/_ ___ , \ . heat of wet steam aftci adia- batic expansion One A\,iy of doing so would be fnsl lo calculate the diyncss q and then apply equation (2) of Art 71, / = /,-]- qL Bui equations (6) and (7) o| that Article give a more con- venient method, which is available here because we know the entiopy of the " Jg "' mixtuie Theseexprcssioiisuia y be(hreell y ol,lauH-(lb V e< ) ns l d ( ., 1 , ) " ' \ what amount of heat the wet uuvhn-e would luue lo ,-, u i,l, l{ weietobewhollycondcnscd,andwhal amouutof heal ,1 ucu.ld | mxi to take up if it weic to be wholly evaporated, urnle, Uu in] THEORY OF THE STEAM-ENGINE 101 pi ess 1 1 re coiiespondmg to the tempeiatuie of saturation Tm eithei case. To bung a mixture at c (fig. 23) into the condition of water at e would require the lemoval of a quantity of heat equal to the area under ec, namely T (<f> - </>J, wheie </> is the entiopy at c and tj> m is the entropy of watei (at e}. On the other hand, to bring it to the condition of satmated steam would requue the addition of a quantity of heat equal to the aiea under cs, namely T (cf) s <). Hence the total heal of the mixtuie at c is / = I w + T (</> - &,), or 1 C = I S - T (<f> s - <). Oi these two cvpic&sions the second is the moie convenient because steam tables gcneially give moie complete sets of values of (/>, than of </> w The entropy cf> of the wet mixture is the constant entiopy under which adiabatic expansion has taken place it is to be calculated fiom the initial conditions This method of finding the total heat ciltci admbatic expansion makes no assumption as to \\hat the btale of live steam was befoie expansion it is equally valid \\hether the steam was diy, wet, or supeiheatcd to begin with What is assumed is that nftci expansion the steam is wet, and that \\i\\ m ooncial be tuie even if tlicic be a large amount of initial supeiheat. IL is also assumed (Ait 78) that the vapoui and liquid in the wet mixtuie aic in cquilibmim In the Knnkme cycle ot Arl 87 it \\as assumed that the steam -uas diy mid saliualcd al the bc'omning oi I he adiabatic expansion Consequently its initial tola! heat was / v and tlnouohout ex- pansion was equal lo </> i Under these conditions the total heat tiflci aduibaLic expansion is /c--^-^(^ B -<W, jnd the hcut-diop is J r i 1 -J 6 = A 1 -A 1 +T.tf. 1 -W To take a numerical example, let the steam be supplied in a dry stiLui.'iLcd stale at a piessiuePi of ISO pounds pel squaic inch, and lei it expand achabatically to a pressuie P 2 of 1 pound pei squaie inch, at which it is condensed. With these data we find from the tables '1\ - 402-58, T 2 = 311 -84, ^ = 1 5620, <f> Sa = 1 9724, y Si - OGS-53, J, t = 012 10 Hence the total heat aftei adiabatic expansion to the assumed pressure of condensation is I = 612 5 - 311 8 (1 9724 - 1-5620) = 484-5. 102 THERMODYNAMICS [en. And the heal-diop, / Sj - / t = 60S 5 - J81 5 = 184 If \\e considci I lie Rankine cycle as a. whole, the feed-pump tcim A (Pi - P z ) V m is (iso-|)i44 xooiGi Q 1400 Deducting this fioin the hcat-diop we have 183-7 pound-calories as the theimal equivalent of the net amount of woik done in Lhe Rankine cycle. The heat supplied is 7 6j _ 2 wt - A (P l - P 2 ) F w> = 668 5 - 38 6 - 30 - 029-6 Hence the efficiency of this Rankine cycle is 629 6 v Tins example will seive, incidentally, to show how unimpoitanL is the feed-pump teim. It i educes the amount of woik done by less than one pait in six hundied II we had left it out of account, and baken the heat-drop in full as the numeiatoi in icckomng the efficiency, the fig me obtained would hn\c been 02923 the cliff ei cnce is insignificant' 1 . A Cainot cycle with the same limits of tcmpeiatuic Mould (Ail 81) have the efficiency 320 The difference between this and 292 shows the loss which i esults in the Rankine cycle fiom not supplying all the heat to the best possible Ihcimoch namic ad^ antagc, namely at the top of the Icmpuatuic lange It amounts in this inslance to not quite 3] pei cent of the whole Heat -supply 90. The Function G. In his steam tables Callcndar gives numeiical values of a iunction G, defined by the equation G =- T<f> - /, which applies to steam in any state, wet, div-satmalcd, supei- heabcd, or supei cooled Bv the help of this function the piocess oi calculating the hcat-diop ma\ be slightly shoitencd G has the impoiLanl propel Ly that it js con-slant dining a pioeess of e\ apoi.i- Lion or condensation al constant pressure Foi in any step of such a * Accordingly <i good approximation to tho c'flioioiic 1 } oi flic R.inkiao cyolo is obtained by leavniy out tho teim A (P a - P 2 )V tl , , in both iuimi.ia.loi and doiionu- natoi of tlio complete oxpiession jn Ait SS, and willing it simply in] THEORY OF THE STEAM-ENGINE 103 process S/ = TS(/> and T is constant, consequently SG = Hence the value of G for a wet mixture at temper at me T and entropy (f>, such as the mixture at c (fig 20) resulting from adiabatic expansion, is the same as G 3 , the (tabulated) value of G for dry-saturated steam at the same pressure Therefore to find I , the total heat of the wet mixture, we have The heal -drop is then determined as before, by subtracting I c fioin the tolal liCtit before expansion. Takino l he same numerical example as m Art 89, T is 311 81, is 1 5020 and G h (for saturated steam at a pressure of 1 pound per square inch) is 2 01 by the tables This gives / - 311 84 x 1 5G20 - 2 61 = 48-1 5, and the heat-drop from the dry-saturated state befoic expansion is 608 5 - 481 5 01 181 as before Oi we may obi am I lie heat-drop even more directly thus, when labulalc'd \alues of 6' aie available The relation / = T<l> - G holds foi any slate ol I he substance Hence between any t\\o poiiilt> (b) and (c) on the same adiabatic line the heat-drop />-/ = (T, - T e ] </, - (G b - G c ) In I lie picscnt example G b is the value of G lot saturated steam a! /'= ISO, which (by Ihc tables) is 51 10. (^ is equal to I he value lor sahualcd slcain al P 1 , \\hich is '2 (>1 The dillcience of tcmpciatuie r l\ Y' c u> 150 71 devices IIoucc I he heat-drop ib 150 71 1 5<>20 -(''5110-2 01) = 181 0, uluch agrees with the rcsull louiid abo\e by less direct methods The use of Hie fund ion G in llns connection is only a matter ol convenience The procedure m Art 89 gives Ihe heat-drop readily ciiouuh, though not quite 1 so shoilly, without the help of G 1 ' * G (with its sign rovoiwed) is one of Lliroo functions to which Willaid Gibbs gave tho ruimo of "Thormodynaimo Potentials" see hit) ticmntijic Papers, vol r llo roprosotifcod llioni by tho symbols ^, x, anc l i" Of thoso, \f/ IH E-T<fi This fiiiiotion was callod by Holmhollz (ho " FIDO Enoigy", it is used in the theory oi solution and otlici applications of theirnodynaimos to chemistry, a subject outsido tlio scope oi this book Tho function x *& tho total heat I, namely E +APV, and is, as wo have soon, of paitioular impoi tance m the thermodynamics of engineering Tho lunotion f is IS-Ttp+APV or I-T(j>, hence 0= -f This function is useful m treating of tho equilibrium of different states oi "phases" of tho same substance One example of auch equilibrium occuis in wet steam, which 104 THERMODYNAMICS [cir. 91. Extension of the Rankme Cycle to Steam supplied in any State. In the Rankine cycle described in Ails 80-87 the steam Mas supplied to the cyhndci in the d^-satuiatcd stale But the teim Rankme cycle is equally applicable whatever be the con- dition ol the woikmg substance on admission, whether wet, dry- saturated, 01 superheated As regaids the action in the cyhndci, all thai is assumed is that the substance is admitted at a constant piessme P l9 is expanded adiabatically to a piessurc P z and is discharged at that pressuie, and that m the piocess thcie is no tiansfei of heat to 01 from the metal, noi airy othei iircvcrsiblc action. In these conditions the heat-diop 111 adiabatic expansion fiom P! to P 2 is the tlieimal equivalent of the nic&jbcL in fig 22 (compare also Art 80) and theiefore nieasines the woik done in the cylinder, no mattei what the condition of the subt, lance on admission ni&y be. This applies to wet s Learn 01 superheated steam just as much as to diy-satuiatcd steam 92. Rankme Cycle with Steam initially Wet A complclc Rankme cycle for steam that is \\et on admission to the cylmdei is shown on the entiopy-tempeiatine diagram by the figuie ab'c'c (fig 24<). The point b' ib placed so that the lalio ab' to ab is a nurture of two "phases, " liquid and vapoui Tlio tuucfciona \j, and j 01 - will bu nici i id to aso-in in Chap VII From the engineering point of view it may bo useful to point out llial tlicso functions liavo the following piojpeity Retailing to Ai( bO, Jig 15, \vo lia\o aotn tliaL vi ken any fluid expands aduibuliudly liom any stato a to any utliei wtato h, tho tliemial equivalent oi the aioa eabj, 01 A\ VdP, is tho hoat diop, l (l I tl , and that Iho aioa tnabn 01 A\PdV is the loas ol intcinal onoigVi ^,,-^1, yjnulniJy, if ab 111 tliat clmgiam repiesent an isothermal piocoas wo Jiavo hvo concaponding pioporfitiona, uith iega,id to the lunctions and \f> When any !Juid o\panda ibothet mally hotu any atato a to auj r ftate b, tho tlioimal oqnualcnt of tho aiua cubf, 01 *Jj ( dP, is G lt -<?/,, and that of tho aioa malm, or A\PdV, IB \jj (l - i/v, To piovo Llus, wo have by definition ^ =E -1 tf> Honco in an isothoimal procose, d^=dE-Td<l> Eut Td<[> it) tho lioat Lakon in, winch IN equal to the gain of mtoaiul onoigy ylus tho woik clone, OL Td>/> = dE-i APdV I 1 ' Tlioiofoio d^--APdV, and $, t ->[>,,= A PilV J Agam, wo have by definition G Tfi-I Ilonoo m an isothoinial prooosa, dG = Td<f>-dI But (by dofuution) I=E-\ APV, horn which Thoi of 01 G dO=-A VdP, and 6 tt -G b =A\' VdP J a in] THEORY OF THE STEAM-ENGINE 105 is equal to q 1 the assumed dryncss on admission The line b'c' represents adiabatic expansion fiom P l to P a , c'e icprescnts condensation at P 2 , and ea re- presents as before the heating of a ^^^^^^^' I he condensed watci The totr,l heat bcfoic acha- /y <; /fV X batic expansion is I Wi + q-jL-^ or / Si (1 g t ) L l and the heat supplied is the excess of this quantity above p lg 24 The cntiopy <j> during adiabatic expansion is <j> Ul + frLJTi or ^-(1-fc) ,/?', The toldl heat aftci adiabatic expansion is I^-T^-fr or T^-G,. The lieat-diop is gol by subtiacting this hom the total Jieat bcfoic achabatic expansion. Or the heat-chop may be iound, as soon as (/> is calculated, by uiing the expulsion (Z\ - T 2 ) - (G, - GJ 'J'he cllici(.it> A\lnch, as bcloie, is praclicall) equal to the heat- diop cln uk'd by Ihc heat suj)phed, is slightly kss than when the slL.iin is saluiattd bclou 1 cxpdnsion, the icason beiuy that the })H)[)(jition ol huit su})plicd at the upjjci hunt ol tcmpeiatmc is no\\ lathi'i less, because pail of the watei icmains uucoiivei ted iiilo sleani As .1 uiiUK'iieal example let g l be 0, and let Ihc lunils of pussuic be Hie same as in Ihc example ol Ait 89 Then the total heat pei Ib of I he mixtiue bcloie expansion, which is / s lL 1) is 008 53 - I X 170 2 = 020 9. The heat supplied is 020 9 38 9 = 5S2 The entropy is The total heat after expansion I Si T z ((f) Sz c/>), 01 T 2 </> G 2 , is 452 4, the heat-chop is theicfoie 168 5, and the same liguie is obtained for it by the duect foimula ( r JL\ T 2 ) (G : G 2 ). Allowing foi the feed-pump term, the efficiency in the complete Rankine cycle is 289, as against 292 when there was no initial tvetness. 106 THERMODYNAMICS [en. In piactice the steam supplied to an engine \voukl be MU only if theie weie condensation in the steam-pipe, such as would occ'in if it weie long or insuflicientl} coveied with non-conduclmg matenal, or if the boilei "pinned " Pinning is a defective bodci action which causes nnevapoiated watei to pass into the slram- pipe along with the vapoui. The above example will show llul i\ modeiate amount of wetness leduces the ideal eflicicncy onl> VCHY slightly, it has no moie than a small effect on the figinc 1 lor I lie Rankine cycle. But its piactical eflect in reducing Llic dlic'ii-ncy of an actual engine is much gieatei, because the picsc-ncc of walri m steam increases the exchanges of heat between it and the nu l.il of the cylinder, and consequently makes the real action dc-parl moie widely fiom the achabatic conditions which are assumed in the ideal opeiatioiib of the Rank me cycle. 93- Rankine Cycle with Steam initially Superheated On the othei hand if .the steam be supciheated before it enleis llu- engine, the exchanges of heat between it and the metal aio mlucnl , the action becomes, moie neaily adiabatic, and the puioim.mcc of the leal engine appioaches moie closely the ideal of I he Rankine cycle Tins is the chief icason why supci healing nnpiovcs I he efficiency of a leal engine of the cylinder and piston typo In steam tin bines it is beneficial paitly foi the same icason and p,u Uy because it i educes mteinal faction in the uoikmg fluid by kec pin" it dner than it would othenuse be during its expansion I In ougli UK- successn e imgs of blacks Supeiheatmg i^ now very gcnt-rairv cm- ployed in steam engmeeimg It is theiefore impoi taut to consult i in some detail the Rankine cjcle foi steam thai is initially supei hc.ilcd In the entropy-tempeiatuie diagiom (fig. 25) the line W iqiic- sents the piocess of snpei heating sLcam that was flry-saliii.il cd nl b. Dining this process its entiopy and its tcmpcratme bolh m- crease, and when the pressuie and tempeiaime at any sta<>(> m the supeiheatmg aie known the coriesponding cnliopy is ft.inul fiom the tables relating to supeiheatccl steam It we assume llial the picture dining superheating is constant, and eq.ml lo llu boilei piessuie, the line bb' is an extension, mlo the ,<., ,,r supeiheat, of the constant-picssure line ab Duiino H K - p.oc.ss of supeiheatmg the .team takes in a supplemciitaiy q.muhly of heat equal to the aiea undei the cuive bb', measmcd dmvn lo (he- base line, namely nbb'n'. This quantity of heat may also be fonnd horn the tables, being equal to the excess of the tolal heat /, Ill] THEORY OF THE STEAM-ENGINE 107 over thai of saturated steam of the same picssuie Callendai's tables give values of the total heat of superheated steam, as well as its en- tropy, iw a wide lange of picssines and tcmpe-iatuies Dining the subsequent pio- ccss ot adiabatic expansion b'c' the steam loses snpei- heat, and il the process is camed so fai lhat the adia- balic line tlnoiigh // ciosses the boundary curve, it be- comes sa Una led and then \vel, and the final condition is lhat of a wet mixline at c' The tolal heal ol this wel imxlnic is found b\ the method aheady dcscnbcd The uoik done in the Uankmc cycle as a vhok. is Ihe aiea eabb'c', and Ihe heal taken in is the aiea i [tithh'n 1 Bolh the.sc quan- tities ,ne ie\i(lily calculated without the help of the chaplain cycle we h,i\e onlj l<> calculate the heat -chop dining adiabatic expansion, namely, I b > - /,', and subtiact fiom that the small In in \\liifh is Ihe Iheunal eqm\ alent of the woik clone m the teeel- pump, namely, / (1\ - P,} /',, The heat supplied is As a niiinc'iic.il example we. may again take Pi = ISO andP, = 1, and assume lhat supciheating is earned so fai as to imse the tem- pei-aluu of the sleam to 400 C , which is a hunt veiy laiely ex- ceeded in piaelice As a uile the tempciatmc aftei supeiheating is eonsuleiably lowei than this. With these data the steam tables show that Ihe total heat of the supciheated steam is 7SO 8 and Us enliopy is 1 7033 The heat-supply is 780 8 - 38 9 = 741 9. Al'lcr achabatic expansion the steam is wet, and its total heat, which is /,, - 2' 2 tf st - 0) 01 T^ - G 2 , is 547 2 The adiabatic heat-drop is therefore 233 G Er<t ropy Fin 25 n n' To luul the \\oik done in the 108 THERMODYNAMICS [OH. Or we can find the heat-chop very directly by help of Callendai's values of G By Art. 90 it is (T/ - T 2 ) - (G/ - G 2 ), \vheie 1\' and GI icfer to the initial state aftei supeihcating. This gives 361 3 X 1 7033 - (406 09 - 2 61) which again is 233 6 If we deduct the small feed-pump term (0 3) the efficiency of the cycle as a whole is 233 3/7-11 9 - 314 This is lathei better than the figure foi satmated steam (0 292) because a poition of the heat is now supplied al a highci tempciatmc. Even with the extreme amount of superheating, howevei, which is assumed in this example, the mam pait of the heat is still supplied at the tem- peiatuie of saturation, and therefoie there is no gieat gam in theoictical efficiency as expiesscd by the ideal Jiguie for the Rankme cycle. The piactical advantage of superheating is much more consideiablc, foi leasons which have already been indicated, than might be expected from this companson 'of the two ideal cycles. In the adiabatic expansion of supeiheated steam a state of saturation is i cached when the pressuie falls to such a value that S foi satmated steam at that pressuie is equal to the entiop}' duimg expansion In the numerical example the entiopy dining expansion is 1 7633, and the tablet, show that this corresponds to satuiation at a piessme of 136 pounds per squaie inch. Any fuithei expansion produces wetness, or else supei saturation If it be desned to tiace the changes of volume dining the adia- batic expansion of supeiheated steam, the initial \olume (conc- spondmg to the assigned piessme and temperature) -\\ill be found in one of the steam tables (see Appendix III) The formula P (V by 3 = constant then applies, down to the picssuie at which the steam becomes satmated In this foi mul a, as was explained in Ait. 78, b is the volume of watei at C , namely, 0160 cubic feet; a teim so small that it can usually be left out Dining furthei expansion, when the steam has become wet, the volume at any stage may be deteimined (as in Ail. 78) by fust calculating the diyness q. Moie cluectly, and veiy exactly, the volume of a wet mixture is found (without calculating q) by Callendar's foimnla (Art. 211) l s Kl whcie K is the minimum specific heat of watei, namely 9967, and t is the tempeiatuie measuied fiom C Since K is veiy nearly in] THEORY OF THE STEAM-ENGINE 109 unity it makes no sensible dilfeience in this formula to write t for Kt and we therefore have 17-17 ^ q ~ ^ r <1 ~ y s r f > J. s b as a convenient means of finding the volume of a wet mixture at any tempciaturc t, when the total heat (7 a ) of the mixture is known. To exemplify these methods of finding the volume we may take the same case as before, namely the expansion of steam at P = ISO pounds per sq. inch and tcmpciatme 100, down to a final picssme of 1 pound pei sq. inch By the tables the initial volume is 3 9605 cub ft We thcicfoie have, dining the fiist pait of the expansion, while the steam slill lelains some superheat, 1 3 log (V - 016) - 1 3 log 3 9145 + log ISO - log P. Tins applies down to P 13 62, the pressure at which saturation is i cached Applying it to that pressure we find V = 28 7, which apices as il should do \\ilh the volume gn en in the table foi satur- aled steam. Assuming that in the subsequent pait of the expan- sion the si cam is in equilibrium, it will be wet, and its volume V ^ is (ound al am stage by (nsl limling / at that stage and applying [he mo'lhod gi\ui above In Ihc final condition, when P 1 and / 51? 2 I lit- volume so calculated is 295 2 cub ft and the div- lU ss is SS(j 94 Reversibility of the Rankme Cycle \Vhatc\ ci the initial slalc be, Mhclhei diy-salmalcd, wet, 01 superheated, the mtiinal achon ol Ihc woiking siibslancc in Ihc Rankme cycle is leviisibli An ideal engine [Kifomung a Rankine cvcle may be legaided .is a slnclly U'\ r i isihk engine taking in heat at various Icmpiiatuies (Ait 1-3), and consequently extracting the gieatest j)f)ssil)le aniounl ol woik out of c.ich clement of the heat supplied, having legaul to I hi tcmpetaluic al which the clement of heat was supplied In the healing of Ihc feed-water a pait of the heat- supply is taken at (cmpciatm.es ranging fiom T, to 1\ But am element ol beat, Liken in at a tempeiatinc T, acts as efficiently as it would do in a Carnot cycle the efficiency of conversion of that T T element is equal to ~ fT - '. Consequently the general efficiency of an ideal engine working on the Rankme cycle is the highest possible ediciency that is compatible with the condition that the substance is to be completely condensed at the lower limit of tcmperatuic and returned to the borlci by a separate pump, instead 110 THERMODYNAMICS I' 1 "- of having its cycle completed by adiabatic compression as in UK- engine of CainoL In othcL- words, the woik which the steam does in the c\ hndri of an ideal Rankinc engine is Lhe greatest amount of uoik thai can conceivably be done by Lhc steam in passing Ihiough am engine., haMng regaid lo the tempera tin c at winch the \M>iknui substance has taken in its heat, and to the icnrpeialuie a I \\hich it icjccLs heal during its complete condensation bcloie bung i<- tinned to the boilci But we know that this uoik is nu.isuml b\ the adiabatic heat-drop Consequently I lit- adiabalic heal-dn>p measures the greatest conceivable pei lonnanre ol UK- sh.un 111 passing through any engine wlrur Ihe conditions ol supph .uul of condensation aic assigned. Whatevei therefore be the nalnic of the 1 engine., the ndiaUitu heat-drop scivcs as an ideal standard with which lo eompait UK actual performance Thus a steam turbine, equally \vilh an < nuim of the cylinder and piston type, cannot exceed, and IK eissaiiK falls short of, the ideal performance- as mensuu-d by lli.il ln.il- drop In the design of steam turbines the ealeulah d valiu <>l I IK adiabatic heat-drop, after making a deduction which is dcl( inniu d b} T experience with similar machines, accoichnglv toiins UK l),i i- on which the dct-iguci estimates the pcifounance lo ht evpcch d In any engine the ratio of the actual amount ol uoik di>n< per Ib. of steam to the amount that would be doiu in UK id al Rankme engine under corresponding condihons ol snppK .ind exhaust, is called the Efficiency Ratio'} . Tests of^ood engiiK s S|K.\\ that in favomablc cases the actual ]K i foimanee is about 70 pi i cent, of the Rankme ideal About 70 pel cent, ol Ihe adml>alu heat-drop is actually converted mlo woik. 95. Conditions of High Efficiency. To seeiin high I'llieu nc\ in the conversion of heal rnLo woik there are obviously lwos pai.id conditions to be aimed at (1) that tlieie shall be n lai^i- IMJI!- drop relatrvcly to the heat of formation of Llu sUam in nlliu words a high value for the ideal elTicicncy, (a) lh,d Iheie shall Ix * To facilitate such ostmmtos tables aio puLhsIiod K'vinfj; (ho hoii(.ilni|) uiuli-i a wide range of itulml conditions as lo ])ICSHUIO and Hupoihoai, mid dual CIHK|I(II.I) as to pressure of condensation Those aio founded on Onlh'iidai'H Kloaiu Tnl.lt << See Heat Dtop Tables, JI Moss (Edwt-rd Arnold, 1917) Tho Htuilciii will liml il n. useful exercine to compare tlio values thoro given with tho honl, ilum an calcMilfifnl by the methods of Arta 89, 90 .ind 93 t See Eeporfc of a Committee on tho Thoimal Eflu-ifiioy of ,St<vuu Mm Pwc In<tt Civ,Eng,\(A oxxxrv, 1898 Ill] THEORY OF THE STEAM-ENGINE 111 a large Efficiency Ratio The second condition depends on piactical features of design with which we aie not at piesent concerned But as regards the ideal efficiency it is impoitant to notice that while some advantage is obtained by increasing the admission pressure, a fai gieatei advantage is obtained by lowering the ex- hausL piessure. That this is so will be clear fiom the following tabulated result^ which relalc to saturated steam The fiist table shows how the heat-diop and the efficiency of Lhe Rankme cycle aie affected by taking diffcienL initial pressmcs, langiug fiom 100 to 300 pounds per squaie inch, but with thii"same piessure of exhaust throughout. Rankinc Cycle f 01 S titillated Steam Effect oj vaujmg the Initial Piessiuc. Initial Heat-chop, to pressure a final pros- ( pounds per wuro of 1 Ih squaio inch, per squnio absolute) inch Woik done per Ih of steam allowing for \\oik spent in feed pump Hoat supplied Ehcipnc\ of per Ih the Rankmo oi steam 03^0 lo 300 (11) CillollCS) 2020 (Ib caionos) 201 5 (Ib calorie^) 034 8 0317 280 1906 199 I 0342 o m 200 1970 1 96 6 033 4 0310 240 1942 1938 032 300 220 191 1 1908 031 7 302 200 JS7 7 1874 030 7 0297 ISO 1810 1837 029 () 0292 ]()() 1798 1795 0283 0280 1 10 17f>0 174 8 026 8 279 120 11)95 1<>93 0251 0271 100 102 9 102 8 023 0201 Kumi llusc icxulls jl will be appaienl that \\hcn (he admission pu'ssuic is high \ r ciy little jinpiovemcnt in tlic efficiency is biought about by even Ji laigT mcicasc 1 of pic-ssuic Tlic twenty-pound use, fo i example, 1'iom 2SO Lo 300 augments the efficiency bv onl} r one pt-i cent On Ihe oilier lumd it is oi gieal tidvanLigc to have what eiigmeeis call i "high vacuum" that is to say to make the piessuie of con- densation as low as possible. If a high vacuum can be maintained and effectively utilized we obtain fiom the steam the work which il is capable of doing under conditions of low picssme but of vciy large volume in the last stages of the expansion The following table illustrates the gam in hcat-diop and m efficiency that results, in the Rankme cycle, from reducing the lower limit of 112 THERMODYNAMICS [cir. pressure. In this example the admission piessure P l is assumed to be 180 pounds per squaie inch, and only P 2 is alteied RatiLme Cycle fo> Salivated Steam Final Effect of varying the Final pressuie (pounds pci squaie inch, absolute) Heat-drop Work done pei fiom an Ib of steam imtialpiessure allowing toi of 180 Ibs work spent m pei square inch feed pump Ho.it bupphed pei Ib oi steam Efficiency of Llie Rankmu cycle (Ib calories) (Ib calones) (Ib. caloaes) 4 144-5 1442 bOl 1 0240 3 153 1 152 8 1 607 5 0251 2 1649 1646 610-1 0267 15 1730 J72-7 621 8 0278 1 184-0 183-7 0290 0-292 0-5 2018 2015 6120 0-314 The last fig me corresponds Lo a vacuum of neaily 29 inches ol meicmy with the barometei at 30 inches. To secure in a teal engine the full benefit of a high vacuum lheslc.ua must continue Lo do useful woik in expanding down to the piessuie at which condensation is to take place In engines of the cylmdei and piston type this is impiacticablc loi two i caserns the volume ol the steam becomes excessive, and the mechanical I'ucLion oi the piston against the cylmdei becomes i datively so gi eat as to dbsoi b all the woik done m the final stages. But with the steam tmbine. these considerations do not apply, theie is then nothing lo piovcnl the steam fiom continuing to do useful woik as it expands ughl down to the piess>me of the condcnsci, and special p.uns ;uc accoidmgly taken to maintain a good vacuum in the condenser of a steam tuibme It is laigely (oi tins icason th.it good sU.un tin bines achieve in piacticc a gicatei cllicicncy lhan even I he besl engines of the cylindci and piston lypc (See AiL. 150 ) 96. Effect of Incomplete Expansion. When steam is ic- leased fiom the cylinder at a picssiuc substantially higher than the piessuie m the condensei its expansion is saiel lo be incom- plete. The effect is to lose available Avoik rcpicscnted by the toe that is cut off the piessmc^ olume diagiam, as in fig 20, and lo make a coiiesponclmg reduction in the clliciency Release takes place at c and the piessure falls to/ while Ihc piston is stationaiy. To exhibit incomplete expansion on Ihc entropy- tcnipern hue diagram, imagine that instead of letting part of the steam escape JIIJ THEORY OF THE STEAM-ENGINE 113 Fig 20 irom the cylinder by opening the exhaust- vah e, -\\e pioducc the same effect within the cylinder itself (as might be done in the engine of Art SI) by applying a icccivcr of heat which will bung the picssuie clown to the lower limit P 2 by causing pait of the contents lo condense bcfoie the piston begins its ictmn stioke The piston being stationai}', the volume of the working substance does not altci dining this pioccss If we imagine the icceivei of heat to have a Lempcratnvc which falls progressively fiom that of the steam al 6- to lhc final tcmpciatuie (T 2 ) at/, this icmoval of heat takes place reveisibls The work done by the steam is not affected by substilulmu this icvcisible piocess for the action of the condenser, because Ihc pussinc in Ihe cyhndci is in no way altcied by the sul)stitulion, but we aie now able to diaw a cm\c that will iipusinl tin puxiss on Hie cnliop\-tcmpciatuic dingiam This is dom in ML> 27, \\hcic the cm \e cf icpiescnts the con- (U nsalion ol pai 1 ol I he sli'.im al constant \ olmm , \\ Ink I IK- piston is al Hsl bc- (01 < hi yummy il s 1 1 1 - I III II si I ()1\C Till I'OII- slanl \ oluini in I his pioci'ss is lo hi- ii ck- onc d pet II) ol steam il is lhc \oluinr ol lhc c\ Imdi i (lixidul by lhc qnanliU of lluid mil m olhri woids il is llu \ohmii pi i Ib ol Ihr \\ct slcam al c C'all lh<il volume l\ Tin n al any li'\ 1 1 of li'inpi'ialmi' such as gill, a poml i on the con- slaiil-volumi' cuive which lepicsenls Ihe pioccss is loiind by lakmg gl ^ p o where V n is lhc volume of 1 Ib of satmalcd steam at lhat tem- peiature. The area of the liguic within the shaded lines rcpicsents Ihe Ihcimal eqmvalenl of the woik clone in the complete cycle The coiner cut off by the curve cf shows what is lost by incom- plete expansion as compared with the work done in a Rankine 114 THERMODYNAMICS . [en. cycle. In the example sketched in fig 27 the initial piessurc (at b) is ISO pounds pci squaie inch, and the steam is icJeascd after adiabalic expansion to 15 pounds pci seniaie inch. 97. Ideal Engine working with No Expansion. If acha- batic expansion weie entnely absent, and steam weie admittcel at Pj dining the whole of the foiwaid stioke of the piston, and dischaiged at P 2 elm ing the backward stioke, the cntiopy-tcm- peiatuie diagram would take the form shown in fig 28, wheie bif is a constant-volume line lepiesentmg the fait of piessuie fiom P! to P 2 This coi i e- sponds, in the ideal cycle, to the action of pinmtive steam- engines such as New- comen's, bcfoie Watt introduced the piac- tice of cutting off the supply of steam at an eaily stage in the stioke and allowing the icma.in.dei of the stioke to be pcifoimed by expansion under falling picssine Points in the cm ve bij aie lonnel as in Ait. 96 In this case the woik done in the cjlmdci, pci Ib of steam, is (P x P 2 ) V^ The net amount of work done, allowing ioi the feed-pump, is (P x - P 2 ) (F % - F Wj ), and the Lhcimal equivalent of this quantity should be equal to the aiea within the shaded lines, of the entiopy-tempeiatme chagiam As a mimciic.il example, assume P t to be ISO pounds pci sq inch, and P l?o be 1 pound per sq. inch, which aie the prcssuics foi which fig 28 has been diawn. Then the woik done per Ib is 179 x 144 x (2 562 010) loot-pounds or 47 17 calories, in companion with the 183 7 caloncs of the Rankme cycle foi the same initial and final pi assures (see the table in Ait 95) The heat supplied per Ib is 020 G caloncs; the effi- ciency oi the ideal engine without expansion is fhcicfoic only 0749 The efficiency of actual pumitive engines woikmg without ex- pansion was much less than this, not only because the pressure \\as less, but because at every stioke a laige part of the steam entcimg the cylmdei became at once condensed upon the walls, and consequently the volume of steam taken fiom the boilci was gieatei than the volume swept thiongh by the piston m] THEORY OF THE STEAM-ENGINE 115 98. Clapeyron's Equation. This name is given to an im- poitant relation between the latent heal of steam 01 any other vapour, the change of volume which it undeigoes in being vapor- ized, and the late at which the saturation piessme varies with the tcmpciatiirc To establish it we may i evert to the ideally perfect steam-engine of Ail 81, m which Carnot's cycle is followed with a liquid and vapom for woikmg substance We saw that this gave an mdicaloi diagiam (fig 10) \\ith two lines of umfoim pressure (isolhermals) connected by two adiabatic cui\es. The heat taken m was L pei unit of woikmg substance, and since the engine \\as icveisible its efficiency was (I\ T)/'l\, fiom which it followed that the woik done, 01 the area of the diagiam, was L (7\ T 2 )jT l This is m thermal units to reduce it to units of woik we multiply by J Now suppose that the engine \\oiks between Iwo tempciatuies which dil'fei by only a veiy small amount We may call the tempciatuies T and T ST, ST. 1 being the small mteival thiough which the engine woiks The above cxpicssion ioi the ^solk done becomes JLS7' T The indicatoi diagiam is now a long nanoM stnp (ilg 20) Its length cib is V ^ - l\ u , V \ being Ihc volume oi unit mass ol the \apom and /", Ihc \ olimu of unil mass ol the liquid Its height is Bl\ \\heicSP is the dillu- - cnce between the picssuie in ab and ^ Ih.il m cd In olhcr woids, since Ihc a \ai)oiu is sal mated m cd as well as in ab, S/ > is the dil'fcicnce in the picssurc oi scihualcd \apoui due lo the dillcr- cnce m Lcmpcratiuc 87" When 8P is lfi made vm small, the aiea of the duigiam becomes moic and moie ncaily equal lo Ihc piocluct of the lenglh by the height, namely S/ J (F, - F w ) Tins is equal to the woik done, whence This equation is only appioxnnate when the interval SP (or ST) is a small finite mtcival In Lhc limit, when the mteival is made indefinitely small, it becomes exact, and may then be written F _ V --- \ 1 ' " w T (dPJs 9 82 116 THERMODYNAMICS |cu. dT where (jn) means thciatcal which I ho lempeiaUne of salmaUd \-i J s vapour changes iclalivcly to the piessure 111 other uoids il is the slope of the satuiation cuive of lempciahne and pussme This is Clapeyron's Equation II may be applied t<> hnd I lie \olinne <>l a satuiated vapom, at any temperature, whi'ii the \olunu ol Hie liquid, the latent heat, and Lhc lale oi' change ol tempt latuie \\ilh presume along the satniation cuivc are known. The values of pi ess nre, volume and LiU'nl lu;il gi\en in steam tables m iclation to tempera Lure, must, if Liu- tables aie .icciu.ili , be such as will satisf}'' this equation Take, loi t \ani pic, sh niu nl 150 C. Callcndai gnes 67-313 pounds per ,s([iiai< mcli loi Hie satuiation piessme at 119 C. and 71 025 .il 151" (' Tin- i.ili at which P is changing pei degice al 150 may be laken .is lull (In difference, or 1856 x 141 pounds per sqnait loot /', is 0017 cubic feet, and the latent heat is 50050 ealmus He net 1>\ Clapeyion's foimula we should find, foi llu s.iluiMlum \oluine in cubic feet, v Annr , 1400 x 50G 5(J V. = 017 -(- - (, "S!) 423 1 x 1 850 X 1 1 1. This agices with the labuLiltd value ((> L',s<)5) We might ha\e obtained Ihe ClajKyion equ.ilion l>\ that the entiopj -tempera I me diani'.un eoiiespoiidmn dicator diagram of fig 20 is a loni- naiiow ship, ^ <j> s - <{> w is L(T and heiyhL is BT Us art a is I be 1 1,, im.,1 , () ,,,\ .,[< ,,| of the woik done, henec SP (F, - rj JL^Tjr, as l, l,. l( In the vaporization of a liquid /" is uinln than / (ll , ,md 1,,-al is taken in, hence by Clapeyron's EquaUon (^ ,s ,,s,l n -. u I,,, h means that mcicasmn Ihe jnessme uuses ||,,. |,,,,| m ,, ,,,, u ,, ( the change of volume V,-V W is known fo, , 1MV s,,bsl.,, l( , ||,,. equation may evidently be used lo lind ll,e ..nu.iml |, x ul, l( |, ,|,, boiling point is uused pei nniL mnease oi pnssn.e 99 Application of Clapeyron's Equation to other Changs m Physical State The leasonmo | )y wlm . h Mlls ( n ' aiiived at was general a applies lo any .eve.sible | 1!U ,, M , he state of aggregation of any subslanee, lf> ,, ,,,,,. } [nm ohdtohquidas wellasto the change Jrom l.qu.d , ^ Th| engine whose mdieatoi diagiam was skdehed in |, 2 ,' I1HIV ,, . anything for working substance, and ,he IM ,H,,M m,l , , J in] THEORY OF THE STEAM-ENGINE 117 fhst opeialion, during winch heat is taken in, may be drawn to icpiesent the change of volume coirespondmg to any change of state In the example already dealt with, the change was fiom liquid to vapom But we might begin with a solid substance pre- viously laisecl to the tempeiature at which it begins to melt (under a given picssure), and diaw the line to icpiesent the change of volume that occuis in melting, while the pressure remains con- stant. The substance does external woik against that pressure if it expands in melting, 01 has woik spent upon it if (like ice) it con- tiacls in melting The steps of the aigument aie not affected, and hence the equation may be wntten thus, with icference to any such tiansfoimation of state, V" V = T dP' where V is the volume of the substance (pci unit of mass) in the fiist state, V" its volume in the second state, A is the heat absorbed in Ihc liansfoimahon, and dT/dP is the late at which the tempera- line of the tiansfoimation (say the melting point 01 boiling point) is uUeiccl by alteimo the piessmc. If a solid body expands in melting V" is gieatei than V and (since the latent heat A is positn e) it follow s that dTfdP is positive in ollici woids the melting point is laised by applying pi ensure On the othoi liand if the body contiacts m meltino V" - V is m-ivitive and dP/dT is motive in othci voids the melting point is Imveied by applying piessine This is the case \\ilh ice. Fiom [lie known amount In which ice contiacts \\hcn it melts, James Thomson (in 1819) applied this method of leasonmg to pi edict thai the mcltiivj> point of ice would be lo\\eied by about 0074 C. j 01 each atmospheic ot picssuie, <md the lesult was afteiwaids veil fled expciimcntally by his biothei, Loid Kelun*. The lowei of the two fixed points used in giaduatmg a thermo- inolei (Ail 15) is the tempeiatme at which ice melts under a picssuie of one ntrnosphcic If this pressme weie lemoved, as it miffht be by putting the ice in a ]Ri exhausted of an by means of * See Kolvm's Mathematical and Physical Papers vol i, p 156 and p 165. The numerical result stated m the text is obtained as follows -A P^dof -water changes its volume in f tewing h om 0100 to 0174 cub ft , and gl ves out SO calories < 1T 00014x273 _ dP~ 80x1400 and if SP be one, atmonpheie or 2160 pounds per sq ft , ST is 2160 x OOO00341 O j 0074 C 118 THERMODYNAMICS \CLI. an air-pump -the tempciature of melting would be raised. The waters apom given off at the melting point has a prcssiue of only 09 pounds pei squaie inch, and consequently if 110 air wcic present, and if the only picssuic wcic LhaL of its own vapom, ice would melt at appioximalcly 0071, C., foi the prelim- would be reduced by neaily one atmospheic The icmpeialme al which ice melts unclei these conditions is called Llie Tt tple PouiL because (in the absence of air) walci-slufl can c:?cisL iL I ha I par- ticulai tempeiatuie in three states, as ice, us water, and as vapoui, m contact with one anothci and in eqmhbunm. 100. Entropy-Temperature Chart of the Properties of Steam. Besides serving to illustiatc the operations of ideal engines, a diagram in which the cooidmatcs aic the en I ropy <nul the tempeiatuie may be used as a gcncial chart foi exhibit mo graphically the piopeities of steam 01 of any oLlier fluid The student will find it instinctive lo chaw foi himself < clisul lot steam, on section paper, to a scale laifto cnoug]! [ O i le.isoiubly accuiate measuiement The general character of such a, cliail will be nppnicnl liom fig. 30 It includes the boundary cmvcs ahcddv tlcsc-i ilxd, uluch icpiesent the i elation of enliopy to Icinpcratniciii snl in .ilt'd ^l( .nil and in watei at the same tcmpciahiicaiul picssmr. 15i-f wccn llu-st- is the \\et legion, wheie Ihc subst.iuoe can CMS I in ctpiilibiium only as a nnxtuie of liquid and vapom Ucyoiul i b( sic am boundary, to the noht, js [he icgion of supc i healed vnpom Now let a system ofLtncs of Con^ftuil J'lCbwuc l>e din vvn I^.ich of these shows the lelation of c/> lo T while I lie snJ>sl,mer cliann<s its state in the mannei of tliL nnaj>mary e\.|)cun iciil ol Ail r>(> Staitmg fiomthe extieme Icll, ahnc ol conslaiil j^rc'ssui e I'm ^.ilei is piactically indistinguishable liom (he boundary C'ni\ f e, sdicllv it lies a httle to the left of Lhatcuive, le.ic'hinn it only wlun (lie tempeiature is such that steam begins lo foi in. TIic'ii il eiosst s Mu wet legion as a hoiizontal sliai^hl IHK-, T lx m^ <-onslnnl diiiinn the conveision of the siibsLance from IKJIIK! mU> vapour Allei reaching the steam boundaiy (he line ol 1 couslaiil pit'ssuie uses rapidly during the pioccss oL' .supcihcaliiiji. Tn Hie fi-imc, live lepresentahve lines of conslanl pressure- are (hawn, n.nnelv I hose foi P = 2, 20, SO, 200 and 500 pounds IKV sciuarc melt. When a sufficient number of such lines aie dia\vn it is easy, by f-iMphie mteipolation, to maik on Ihc clmil Iho position ol a })oiul Ill] THEORY OF THE STEAM-ENGINE 119 coi responding to any assigned condition of the substance as to picssmc and tcmpeiatnre, and to tiace, by measurement instead of by calculation, the changes which ensue dining adiabatic expansion. The convenience of the chait foi such purposes is increased by including a system of Lines of Constant Total Heat. Examples of these lines aie shown in Pig 30, for each intei val of 25 calones fiom 1*800 400 350 300 075 1 50 I \ . I 10 I 25 Enuoov Vi<f '!(> Kriliopy Li mpoiiiluio (Jli.u I Im W.iltu ,iud ( Sl 20 1 25 / = GOO to / =. SOO c.iloncs The} aie si)ccicilly useful in I he icgion of supeihodl, llic}. m,i\ ho\ve\ei be diawn in the \\eL 10151011 also As nil example Lhe line foi / = 050 is conLmiic-d inLo Lhc \\cL legion, iL undeitiocs a sluii[) eliange of dueeLion in crossing Lhc slciim boundaiy Each of lliesc lines icpiescnts whaL occius in a Ihrollhng process The lines of constant toUil heat tend, at Hie extieme righl, to become neaily straight lines of constant tempera- ture this is because the vapoui behaves more neaily like a pericct 120 THERMODYNAMICS [en gas the moie the pressure is i educed In a pcifecl gas, as we saw in Ait. 19, T is constant when the expansion i^ of such a nature as to keep / constant. Another useful addition ih a set of Lines of Con&tanl Dii/ness in the wet ieg ion. These aie diaun in the nguie foi <y = 1, 2, 3, 4, 5, 6, 7, 8, and 9 They divide each honzontal width between the two boundaiy curves urlo equal paits (see Ait SI) Lines of Constant Volume may also be drawn in the mannci aheadv descubed (Ait 96) With the aid of such a chait one may find, foi example, by di awing the appiopnatc adiabatic (yeitical) line, that steam wilh an initial piessine of 200 pounds pci squaie inch, superheated lo 400 C., becomes satin atcd when its picssmc falls, by adinbal it- expansion, to 16 pounds. Continued into the wet legion the adia- batic cuts the constant chyness line q = 9 at 50 C , showing lhal theie is 10 per cent of walei present when the piessmc has fallen to 1 8 pounds Theheat-diop may be mleiicd, but for its mcasuic- ment a bcttci foim of chait is one which will be dcscnbcd in the next Article By chawing a \ertical line to lepresent the adiabatic expansion of a mixtuie of steam and watci, it is easy to tiace Ihc changes that occur in the piopoition of walcr to steam In the icoion ol oidmary woikmg piessuies the line foi g = 5 is neaily veitieal. Hence if theie is about 50 pci cent of water picscnt at the begin- ning oi achabatjc expansion, neaily the same pcicentage will be found as the expansion goes on When the steam is much wcltci than this to begin with, adiabatic expansion makes it clnei. In fig. 30 a con]cctuial cmve has been added (shown by i bioken line) connecting the watci and steam boundaiy cmvcs in the legion ol high piessme, whcie, at present, Ihcic aic no dal.i foi a pi ease dctcimination of the cutiopy Tins bioken line is simply a smoolh cnne foiming a eontmuahon of each boundaiy cuive, and diawn so thai it (ouches Ihc isotheimal foi 305 C , lhal being the cntical U'lnpcialnrc foi walei It is a I the culic-al lem- peratuic thai the distmclion between <, and ro disappcais The- honzontal mlciccpt between the water and steam boundaiy cuives, which coiicsponds lo Ihc (akmg in of latent heat, Iheic vanishes the culical point is Iheiefoic at the summit of I ho T<j> cinvc At sufficiently high piessuies Ihc lines oi constant piessmc would pass, in the foim of continuous cmvcs, clear of the lounded top, fiom the legion of walei to lhal of snpei healed steam. in] THEORY OF THE STEAM-ENGINE 121 The water boundaiy cmve is concave on the left for tempeia- tiues below 250 C., because the use of entropy pei clegiee, which is rr/T, wheie a it, the specific heat of watei, becomes less as T in- creases, a being ncaily constant at low temperatures But afc highei tcmperatiu cs the specific heat of watei increases so fast as to make a/T mciease with using lempeiatuie the curve accoid- ingly bends ovci to the light a-, it appioachcs the cntical point. In the next chapter we shall have occasion to icfei to examples of cntiopy-tcmpcialme cluuts foi other fluids In one of these caibonic acid the. icgion which is practically impoitant, in con- nection with icfiigciating piocesses, includes the lounded top whose summit is the critical point In that chagiam the lines of constant piessme in the liquid aic cleailj distinguishable fiom the bounddiy curve ol the liquid slate 101 Molher's Chart of Entropy and Total Heat While the cntiopy-tcmpciatiue diagiam is invaluable as a means of exhibiting thcnnoclMiamic cycles and as a help towaids undci- standmg them, another diagiam, mtioduced in 1901 by Di R Mollici ', is of gicalci senicc in the solulion of practical pioblems By taking I'oi cooidmatcs the cntiopy and the total heat, Molhci consliucls a cluul which fiom tins point of MOW has advantages thai cnlillc it lo the fust place among dc\ ices for lepicscntmg giaphiCiilly I he Ihcimodynannc action of steam in steam-engines, 01 of the woikiug fluids in iefi igeialing machines Its applica- lions m icfngciahon \\ill l)e dealt uilh in Ihc next ehaplti As icgaids steam il fuinishes Ihe mosl convenient \\a^ to mcasme Ihe heat-chop m tididbatic expansion, \vhatc\ci be the initial slale <is lo snpciheal, and conscqncntlv lo find Ihc gicalcsl lluoicheal oulpul thai is allamable when Ibc inihal picssmc and U mpc'ialmc, and Ihe final piessuic, aic assigned We 1m e seen I hat tins can be calculated when tables as complete as Callendai's aie available, and also that il can be found by the aid of an cnliopy-lcmpeialiiic chait on which lines of constant total heat have been diawn Bui the Mollier chait allows a giapluc solution lo be obtained with gicat du eel ness and case Foi practical purposes the Mollici 1$ chait is diawn so as to show only the steam boundaiv curve and the icgion immediately above and be-low it, but il is mshuclivc lo considei the complete * Jl Alollioi, "Ncuc Diagiamme zur lochnischon Waimclciluo," ZahrJnift (lea Veremea dcuhcher Inycmeure, 1904, p 271 See also IHR Nnie Tabellen und Dingrammz flir Wasaerdampf, Boriin (Jubus Spnngci), 1906 122 THERMODYNAMICS [en chart for watei and steam, which is sketched in skeleton foim and to a veiy small scale in fig 31 Theie ea is the watei boundary curve and bs is the steam boundaiy cuive The straight lines between them, such as ab and es, aie constant-pressuic lines, one of these (for P = 200 pounds per sq. inch) is continued across Lhc boundaiy into the region o bupeiheat, the ciuvc bb' icpieseiits the piocess of supeihcatmo at that pressure. The slope of smy line of constant piessure is a measiue of the tempeiatuie, for at con- 800 Fig 31 Molbci I(j> Cluil for WaLor and Sloam 20 btant pressuie dl = cZQ = 2V%and consequently rf//<7r/> = T In the wet legion the tcmpciatme along any line of constant picssme is constant, being the tempciatuic of satmation foi that piessurc, and therefore any constant-pi cssme line in that region is straight It ciosses the steam boundaiy without change of slope, but gradually bends upwaicls m the icgion of superheat as the temperature uses, foi its blope continues to be a measure of the tempeiatuie. The \vatei and steam boundaries aic connected, as in fig. 30, by a conjectural line bhiough the cutical point. The cnlieal point mj THEORY OF THE STEAM-ENGINE 125 is not at the summit of this line, but at its point of mflectioif, which is also its point of maximum slope At the cutical point the continuous boundary curve, shown by Lhc broken line, would touch a cuive of constant pi ensure, and consequently its slope there, dl/d<^, is equal to the critical temperatuie, the absolute value of which is G3S The broken line is accordingly drawn to have a slope of 038 units of / for 1 unit of cf) at its steepest pait wheic, foi some distance, it is very ncaily stiaight*. Each constant-pi cssure line in the wet region may have its length between the two boundary curves divided into parts which express the dryncss q at successive stages in the pioccss of vapori- zation, ]ust as in the T(j> chait. Since the heat taken in up to any stage of that process is pioportional to g (Ait 70), equal distances along the line, coiiespondmg as the} do to equal mciements of total heat, coiicspond to equal changes of chyncss In this way lines of constant diyness aie dctcimmcd, sonic of which die shown in the sketch. It is useful to ha\ c a system oi lines of constant tcmpeiatnie diMMn in the icgion ol'snpeihcat two such lines aic shoun in fig 31. When they and I he constant-pie^siuc linos m thai legion have been chawn it is casv to maik Hit 1 point which corresponds to any assigned condition of the steam as to tcmpciatuie anel piessme. Thus b' is the 1 j)omt coiie'sponclmg to steam uilh a piessme of 200 pounds, siipcihe^atcel lo 100 C 1 Then by dialing a vcitical sliaighl line thiongh the- point so found, \\e^ exhibit the pioccss of adiabatic expansion The lenglh ol Ihal line, elo\\n lo the final picssmc, nieasiue-s Ihc adiabalic heal-diop, and Ihcicfoic gncs a vciy simple and dncct means of finding Ihc gicalcst amonnl of \\oik ideally obtainable. horn a poiincl ol Ihe woikmg substance Thus I he 1 heal-diop in adiabalic expansion down lo a picssure of one pound per sqnaic inch is determined b> measuimg b'c on Ihe scale of / The position of c among Ihe- lines of constant di3iicss * Tlio Rlopc of tho boundaiy ciuvo, which is ( ) , w equal, at Uio critical point, \ " 7 ' I s to tlio slopo of the connUnl-prossuio lino which Luuc Iios li Uioio, nanu ly f J But ) = T, mnco m any constant prossiuo clian^c dT = Td(/> Honr o at tlio critical point ( f 7 - ) -- T Houco also, at that pomt, ( 7 ) = ( ) But ( , ] .which 1 U0/H \&p-Jt \^L \dl>J ri is the slope of the boundary curve m the enkupy-tompoiatmo chatt (fig 30), is zoio at the critical point Ilonce at the ontioal point , , J =0, tliat is to say the boundary curve in the /</> chart theio undoigoes inflection 122 THERMODYNAMICS cJnows how much of the steam is condensed by llus adKibaiie expansion. The advantage of a high vacuum, Lo which attention was diawn m AiL. 95, will he obvious fiom llic cflceb of the fin.il pressiue on the length of b'c. A thiotthng. f pioccs,s is represented by a hoiuonlal sliaii>hl line, suice / is constant Lines of consLant tcmpcialmc in I he supei- heated legion become neailv stiaight and liomontal aL very low O 1,1 piessuies, for the behiu loin of the vapoui then appioxunates lo that of a peifect gas. 800 ___________ ~40QC . --- 750 =Q 7 EntfOfjy (p 15 16 \T \ 8 "l 9~ 32 Mojhd's Cliari of Total JTcuil and E 20 22 A complete Rankmc cycle is shown by Ihc closed ligine cabb'cc, wheie ea is the heating oi the feed-watci, ab Ihefommdon ol slcain in the boiler, W its snbsequenL snpcrhcalmg, b'c iL^, adiabahc expansion to the piessuic of Lhe condcnsci, and cc ils, coiuhnsalion at that pipssure Foi the piaclical use of the diagram, ho\vcvei, theic ib no need to include the whole cycle Whal is wanted is the legion to the light, wneic the quality of the slc.uu IK fore and after expansion is exhibited, especially [he u-gion ['mm m] THEORY OF THE STEAM-ENGINE 125 /> - 1 5 to 2 and from I = 450 to 800, and by restricting the uhail Lo this i eg ion open scales may be used without making it unduly laige. Fig. 32 gives, in mmiatmc foim, a Mollier Chait for the useful region, showing, a few lines of constant piessure, also lines of constant Icmpeiature in the legion of superheat, and lines of constant dryness> in the wet legion*. 102 Other Forms of Chart. Besides the foregoing diagiam Molliei mil educed anothci in which the cooidmates are the 800 100 Fig 33 Pi esiuni Pounds per ,y inch 200 ' 300 400 "TOO Mollic'i's (.'luut of Total Heat and PUHSUIL picssme and the total heat A skeleton PI chait foi steam is shown in fig. 33 foi the icgion useful in piacticc. It has I he piopcity that lines of constant Lcmpeiatuie and lines of constant volume aie straight. It includes the steam boundaiy curve and a paitof the wet icgion below it, which is mapped out by lines of constant diyness as in the other chatts Tluough the wet region and the legion of * A chart of t,hi kind, exlubitmg Callenrlar's figuies on a scale large enough for practical use, has been diawn by Piof Dalby for his book on Stoarn Power (E Arnold) and may be purchased separately 126 THERMODYNAMICS [en. superheat above, lines of constant volume aie diawn. They arc stiaight in the region of supeihcab, and sensibly stiaighL in the wet region, but they undergo an abiupt change of direction on crossing the boundaiy. (See Aits 208 and 209 ) Vaiious other chaits may be deviled by selecting for the two cooidmates othci pans of propeities fiom the list given in Ait 75. In any such chait the characteristics of the fluid arc exhibited by drawing systems of cim es, each of which icpiesenls the iclaLiou that holds between the two pioperties chosen foi cooidmatcs when the state alteis in such a manner that some thud property is kept constant By chawing seveial such systems of curves a compio hensive giaphical substitute for numeiical tables may be con- structed. The particular piopeities selected foi the coordinates, and for the cmves, may depend on the type of pi obi em 01 problems foi which the chait is wanted. Callcndar gives, as an adjunct to his steam tables, a chait m which the cooidmates arc the total heat and the loganthm of the piessmc. With lespect to all such devices it may be said that, so far as steam is concerned, the publication of full tables, which include the region of supeiheat, lender giaphic tabulation less necessaiy. It is now comparatively easy to find any icqmicd quantities chrectlv fiom the tables, 01 by mteipolation fiom them, with gieatei accuiacy than is leached in measuimg fiom a, chait But foi ceitam pm poses the giaphic piocess it, sufficicntlj'- exacl and more convenient All students should in any case make thcmseh cs acquainted with the entiopy-tempeia,Luie chait, and also with (he Molhei chait of entiopy and total heat the foimer because it will help them to undcistand cyclic pioccsics, the lallci as an insliu- ment for dealing with piactical pioblfins m steam cMgmccung and mechanical lefngeiation 103. Effects of Throttling. We have ahcady .seen (Ail 72) that when a tluottlmg piocess is cained out nuclei conditions lh.it pi event heat fiom cnteung or leaving the substance the lolal heat / does not change. Lines of constant lolal heat on any of the diagiams accordingly show the changes in othci qiitiiililies which aie brought about by thiottlmg. It JS the piocess that occurs when a fluid passes through a "icducmg valve" 01 other constricted ounce such as the porous plug of the Joule -Thomt,on expeiiment (Ait 19) It is not what occurs when a jet is foi mod, as in the nozzle of steam tin bines In that piocess, which will bo rr] THEORY OF THE STEAM-ENGINE 127 Icalt ^ith latei, the stieam of vapoin acqunes kinetic eneigy that nay be tinned to useful account, wheieas in thiottlmg, any inctic eneigy acquiied in passing through the constriction is mmcdiately dissipated by internal fuction. In a perfect gas throttling produces no change of temperatnie Ait 19), but in steam and other vapouis it produces a cooling ffcct which is measured as the fall of temperature pei unit fall >f pressuic under the condition that I is constant, or Cooling effect = ( - -) . \dPJx Values of this quantity for steam nuclei various conditions can >c deduced from Calendar's tables In steam that is highly o m/ upeiheated, especially at low picssuie, it is small, foi the con- lition of the steam then appioaches that of a peifect gas, but if he steam is saluialccl or only slightly superheated the cooling ffcct of thiottlmg is much gi eater Thus with steam at a pressme if 20 pounds per sqnaic inch, the cooling effect is only 0513 at 00 C but is 338 at the tcmperatme of saturation These aic he falls of Icmpciatuic, due to thiottlmg, foi a chop in piessuie f one pound pci squaic inch The cooling clfect plays an un- (ort.mt pail in dclci mining the values of the total heal and othci aopcitics of ilic vapom, in the method used by CaHdidar 1 " the values given m his tables foi the total heat of * Callundar tabulates for steam a quantity (called by him >C) which is the >roduot of tho cooling eftcct and tlio specific heat at constant pies=uue It is a uniiUl}' of heat, namely the nuinboi ot ealonef) which would lu\e to be civ en D each Ib of tho thiottled steam to rostoio it, at constant pie-^me, to the tompeia uro it had before throttling, when tho amount of (hrottlmg is such that the piossuro f/7'\ tops hy one pound poi squaie inch Tt may bo written /> 01 K,, ( 7P ) it Js equal to \tU 1 1 - ( ) and is independent of tho piossuro (as will bo shown later) The values of SO" or p which aio given in the (able for saturated steam theiofore apply alao o mipeihcntod steam at tho same temperature The cooling olToct C may bo found by dividing tho tabulated values of "SO" by ho specific hoat Tho specific hoat, which is ( - ) , changes only slowly with the omporature It may thcioforo bo found from the tables, for any given pressure nd terupoi atu ro, by noting tho difference between values of I at that pressure and t tompeiatnros above and bolow tho given tompoiature, and taking the amount by rhioh 7 olmngoE por degree Thus, for example, at a constant pressure of 20 Ibs ho rate at winch 7 changes with tho temperature is 509 calory por degree m lio neighbourhood of saturation For saturated &team of that pleasure "SC" is ivon as 172, hence tho cooling effect of throttling, por pound drop of pressure, is 172/0 509 or 338, as has boon stated in the text 128 THERMODYNAMICS [en. supeiheated slcam, it is easy to calculate how much the steam is cooled by any given drop of picssuie in throttling Let saluuited steam, t'oi example, at 200 pounds per squaie inch be throLLlcd down to a piet,suie of 20 pounds. The value of /, which icmains constant, is 609 T. At 20 pounds, the table shows that this value coiiesponds to a temperatuie of 163 8 The saturation tempcia- tme for 20 pounds is 108 9. The ongmnl tempera tin c was 19-A 3 Thiotthng has theiefoie cooled the steam by 30 5, but at the same tune it has caused it to become supci heated to the extent of 54 9. The apparent paiaclox, that thiottlmg both cools a vapoui and supeiheats it, is due to the fact that when the pressure is reduced by throttling the saturation tempeiatiue has fallen more than the actual temperatuie has fallen. Hence satuiatcd steam is supci- heated by thiottlmg, and steam that is initially superheated becomes moie supei heated. Similaily, a mixtuie of vapour and liquid is paitially dned by throttling, it may be completely dncd and even supeiheatcd if there is not much initial wetness and if theie is a sufficient piessuie-drop This is illustiated m iig 30 by the line of constant total heat foi / = G50, which is cliawu paitly in the wet legion and paitly beyond it It bhows the cifccl of thiolthng on a wet mixtuie that contains (J S pci cent of water a! a press me of 500 pounds, the steam becomes chy when the pict,Mire is icduced to 37 pounds, coiiesponchng to the tcnipeiatuic of 128 at which the line ciost.es the satuiation cui\ r coi steam boundaiy The piocess of throttling is still moie simply shown by hoir/ontal lines (/ = constant) in the Molher diagicim (fig 31) By thawing such lines tlnough the points on the satuiation ciuvc for 2' 1 and P = 15 it will be t>een that 12 per cent ol watei can bo 10- moved fiom steam at 200 pounds pressmc by throttling il down to 1 pound, 01 fully G per cent, by thiottlmg it to atmospheiic piessme. Smnlaily, it is easy to liacc the extent lo winch liquid will cvapoiaie in escaping tlnough a thro I tic-valve fioin a i eg ion of high pressure to a icgion of lowci picssuic. The method of chymg by thiottlmg has been applied as a means of determining the peicentage of watci picscnt in steam For I Ins puipose a device is ui,ed that is called, lather nuppropnalrly, a "thiottlmg calorimeter." Its essential feature is a pipe through which a sample of the steam to be tested can be passed, containing within it a diaphiagm with a pin-hole oiifiec, 01 a tluottle-valvc 01 poious plug, tlnough which the steam has to pass. There are presbine-gauges on both sides, and a theimometei to read the nij THEORY OF THE STEAM-ENGINE 129 temperatuie of the steam immediately after passing the obstruc- tion. Both parts of the pipe must be thermally insulated, so that no heat is lost, nor conveyed by conduction fiom one part to the othei The amount of steam passing, which may be regulated by means of another valve beyond the obstiuction, should be such that the steam aflei tlnottlmg is appieciably supeiheated, in oider that no wetness may be lefl in it, complete dnness is ensuied by seeing that the tempciatuie aftei throttling is someAUiat highei than the satmation tempciatuie Then fiom the tables we find I' the lotal heat which concsponds to the tempeiatme and piessuie as observed after tlnottlmg Since thcie has been no change m the total heat, this must be equal to I w + gL, wheie these quan- tities icfei to the state befoie tlnottlmg Hence the initial chyness is found, namely r> r T' r ' J _ *_ _~ l w _ - 1 ~ J-w Q ~ T. ~ T T ' Ms J. ^ J. w In practical applications of this method a poious ping is to be pu'lcncd to a thiotllc-'v al\e because Lhe thcimomctei can be placed close to it and the lempciattue mcasuied aftci the thioltlecl sticam has lost its kinetic uicigs and bcloie it has stilfcicd loss ol heat It is dilncull in am case to scenic that the sample tested by any such appaialns shall be piopulv icpiesenta- h\c, in icspcct of the nioistine it caincs, and consequent^ little leliancc c<m be placed on tests thai me earned out b\ duelling ,i poil ion of a steam supply into a thioltlmi> caloi nuclei, as a means of deleimming the gcncial \\clncss of the supply. 104 The Heat-Account m a Real Process The pioccsses which luiAc been consideicd in lhischai)lei as going on in a steam- engine aic ideal in the sense that they have been assumed to be a (hail ic) Dial that is to say, thcie is no tiansnnssiou of heat to or fiom the woikmg substance except what is oiigmally taken fiom the source 01 finally iqccted to the icceivei , m all the mteimeehate opt rations the working substance has been enclosed in vessels that aie assumed to transmit no heat. The assumed piocesses arc also ideal in the sense that they are internally reveisible The piocess of throttling, which is a typically mcversiblc piocess, did not occur in the ideal engine cycles In dealing with it also, howevci, we postulated adiathermal conditions, it was assumed in the argument of Ait 72 that no heat passed by conduction thiough the con- taining walls to or fiorn the space outside. 9 B T 180 THERMODYNAMICS [<". Discarding these limitations we may now diaw up, 111 general teims, a balance-bheet or heat-account foi any real piocess, winch will include thermal loss to the space outside and also meuTsible actions within the engine 01 other apparatus Whethei the apparatus considered be an engine cyluulc'i, 01 I he- series of cylinders of a compound engine, 01 a tuibinc, or a I In oil - ling device, we may in all cases compaic UK- stale of Ihc Hind Jil entiy and at exit, as foi example in the admission pipe- of an engine and in the exhaust pipe We imagine a steady (low ol' llu- woi king fluid thiough the apparatus At entiy let ils pubsuie be* P { , ils volume (pei Ib.) F x , and its mteinal eneigy 7i' L At exit Id. ils pressure be P 2 , its volume F 3 and its internal encigy A' 2 . To nuiUc the companson complete we may wnte K I'oi Ihc kinetic energy (also per Ib.) of the stieam as it enteis, and A' 2 foi ils knulic energy as it leases In passing thiough the apptuatus flu- Jluid will, in geneial, do exteinal woik, and also lose by conduction some heat to exteinal space. Let W rcpicscnt, in thc-imal uiuls, this output of woik, and let Q L rcpiesent the heat lost by conduc- tion to exteinal space, both of these qiuin Lilies (like Ihc olluis) being leckoned pei pound of the Jluid that passes thiough Each pound that enteis the appaiatns rcpiesenls a sii|)ply of energy equal to K t + E l + AP^V^ , 01 E l is the internal cncigy il cames, and PJ /r l is the woik done by the fluid behind in pushing it in. But E-L + AP 1 V 1 is equal to I l} the total heal pc-i pound ol the fluid m its actual state at cntiy Sinulaily, each pound lhal leaves theappaiatus icpiesents a i ejection of cnejgy amounting lo K z + E, + APjFz, foi E 2 is the mteinal cneigy which (In Hind cames out, and P,F 2 is the woik spent upon il by lh< Iliud lu-lniid in pushing it out E, + AP 2 V Z is cqu.il lo /_>, (he lol.il he;il p< i pound of the fluid m its actual stale at exit IK nee, by Ihc con- seivation of eneigy, foi the apparatus as a whole, /L! + /i = /iT 2 + / 2 -I- W + Qi The terms on the left of tins equation icpiescnl the eneigy (h;U enters the apparatus, the tcims on the right show how il is disposed of in the issuing stieam, in output of uselul woik, and in lenluige of heat The teims K t and K z are usually veiy small, except when (he apparatus is one foi foimmg a steam jet, in which case A'., is (he useful teim this will be considered m a lalcr chapter When the change of kinetic eneigy m the stream is piactically negligible, uij THEORY OF THE STEAM-ENGINE 131 as it is between the admission pipe and exhaust pipe of an engine, wehave Ii-Ii+W+Qt. And when, in addition, the appaiatus does not allow any appieci- ablc amount of heat to escape to the outside (Q t = 0), we have Thus means that when Iheie is a stead}' flow ot a Avoiking sub- stance through an> theimodynanuc appaiatus, the output of woik is measmcd by the actual Hcat-Diop, ivhelhei the internal action is 0) 'is not levasible* piovided theie is no loss oi heal to the outside by conduction thiough the walls The actual heat-chop must not be confused with the achabatic heat-chop, which is Lhe diffeiencc between l and that value which the total heat would reach if theie were achabatic expansion to the exit prcssuie P z The actual hcat-diop I : /, is identical with the achabatic hcal-diop only when there is no loss of heat to the outside and when, in addition, the internal action is wholly leveisible Any uicveisiblc fcatinc in the mtcinal action will mciease /, abo\ e the value which \\onld be i cached b\ achabatic expansion, iind will consequently diminish Lhe output of woik In the cxlicme case of a lluotthng piocess Ihcic is no output ol woik, and theicfoic J> = f 1 , pio\ ided theie is no lo-,s ol heat to the outside Am loss ol heat lo the outside in a thiollhng piocess will make /., coiicspondmgK less foi we then lia\ c /, = / x Q, The losses ol theimod\ nanue cllecl in a util engine which make ll'lcss lhan the ideal output, namch the \ aluc coiKxpondmg to the .uliabalic he.il-diop, ansc paith horn loss of heat lo the outside and pailh horn I wo kinds of uieveisible mteinal action One of Ihese two kinds is mechanical, the othei is thcimal In the mechanical kind, I he acLion mvoh cs (hud hiction within the woilung subslanee It is ol Ihe same natuie as that which occuis in Ihrotlhng llieic is uie\ cisiblc passage of the uoiking substance fiom one pait ol the engine to anothci whcie the picssuic is lo^ei, as foi instance the passage of si cam thiough somewhat eonstiictcd openings into the cyhndci, 01 its passage, on icleasc aftci incom- plete expansion, into the exhaust pipe, with a sudden diop of pres- smc or again, there is the same kind of ineveisibility in a tin bine in the fnctional losses that attend the formation of steam jets or in the friction of the jets on the turbine blades These are all instances of mechanical meversibility. In the second kind of 92 132 THERMODYNAMICS [c-ii. ni ureversible acLion there is exchange of heal bclwccu (he working substance and the internal suilace of Lhc engine walls. The hoi steam, on admission to a cyhndei which has, jusL been vacated by a less hot mixtiue of steam and water, finds llu> surfaces coldci than itself. A pait of it is accordingly condensed on I hem, whu'li ie- evapoiates after the prcssuiu has fallen through expansion. This alternate condensation and ic-evaporation involves ,i consult rable deposit and iccoxci}' of heat in a manner Ihal is not re \cisible, foi it takes place b} contact between ilmd and mcl.d al dilUicnl tempeiatuies. The action may oeciu withoul loss of lii'al It) the outside it would occui, for instance, in an engine wil h a, ceinducl mg cyhndei covered extcmalty with a "lagging" of non-conducting matenal. Its effect, like that ot thioltling 01 lluid Jnclion genci- ally, is to leduce the output of work below Ihe limit Ihal is attainable only in a le^-eisible piocess, and it does this by making the actual heat-chop I I / 2 ^ css Ihan the achabalic heal-diop The equation W = I I z takes account of both kinds of uie- veisibihty of the effect of theimal exchanges wi I Inn I he appai.ilus, as well as of any throttling or fuchonal effects in I he aetion of I lie woikmg substance But it docs not take aeconnl of heal losl lo the outside, and foi that the teim Qz has to be deducted, m, living W = ll ~ /2 - l The full statement of the heat-account in a teal piocess may be expiessed as follows When thcic is a steady (low of a woikmg substance thiongh any thermodynamie appaialiis, the out put ol woik ib measuied by the actual heat-chop fiom c-nlMiiee lo (\il, less any heat that escapes by conduction lo Ihe outside, and h ss any gain of kinetic eneigy of the issuing sticam ovei (he c nUim<r stieam, 01, in symbols all these quantities being expiessed in Iheimal nnils, and reckoned per unit quantity of the woikmg substance. This equation also applies to reversed heat-engines, or he.il- pumps, which will be consideied in the next chapter, but in Hum the quantity W is negative- work is expended on the machine instead of being produced by the machine. In such machines Q, is also geneially negative, for as a rule the apparatus is colder Ihan its surroundings and the leakage of heat is imvards CHAPTER IV THEORY OF REFRIGERATION 105. The Refrigeration Process. Refrigeration is the le- moval of heal from a body that is colder than its surioundmgs In cold stoiage, foi example, the contents of a chamber aie kept at a tcmperdtuie lower than that of the an outside, by extiactmg the heat which continuously leaks in thiough the imperfectly in- sulating walls, To maintain a icfi iterating pioccss lequnes ex- pendituie of eneigy It is generally done by means of a mechani- cally dnven heat-pump, working on what is essentially a icversed heat-engine cycle. It may also be done by the direct use of lngh- temperatuie heat without mtcimcdiatc conversion of that heat into woik We shall consiclci Jatci the duect application of heat to effect icfngeiation, but shall in the fust instance tieat of re- tiigciatmg machines dm en by the expendituie ol mechanical powei. Any pioeess of refngci Alton imohes the use of a woiking substance which can he made to take in heat at a low tempeiatuie and dischaigc heal at a highct temperature Tlic heat is dischaiged by bcjng <>i\ en up to the an outside 01 to any walei that is available to icceive il The piocrss is a pumpmg-up ol heal tioin Ihc level ol teinpcialme ol the 1 cold bod\, at \\lnch it must be taken in, to the lexcl ,1,1 \\hich it max be dischaiged These levels should be as ncai toyclbei as is piacticablc, in oidct that no unucccssaiy woik may be- done in ol her uouls the action of the woiking substance should be confined to the nauowcst possible lange of tcmpcratuic The U'lnpeiatnic of dischaigc should be no highci than is necessaiy to ncl lid of the heat, and the kwu limit should be no lowei than wtJI ensure Iransfei of heal into the icfiigciahng substance fiom the cold boety whose heat is to be extracted Let r l\ be the tcmpciatiuc at which heat is dischaiged and T 2 the tempcuiluic at which it is taken in fiom the colel boely Con- sielei a complete cycle in the action of the woiking substance. Let QJL be the quantity of heat which is discharged and Q 2 the quantity which is taken in fiom the cold body; and let W be the thermal THERMODYNAMICS [en. equivalent of the woik spent in diivmg the lefngcialing machine. Then, by the conservation of eneigy, The useful lefngeiatuio effect is mcasmcd bv Q ? , and the Ck co- efficient of peiiormance," which is the latio of that effect lo the Q woik spent in accomplishing it (Art 4) is , ~ . 106 Reversible Refrigerating Machine. We have first lo cu- qune what is the highest possible coefficient oJ peifoimancc when the limits of tempeiatme 1\ and T 2 aic assigned We know by the pimciple of Carnot (Aits 33, 39) thai when heat passes down from Tj to T , thiough a heat-engine, the ideally greatest clliciuicy in the conversion of heat into woik is obtained when I he engine is thermodynamically reveisible In that case The output of woik W is Q 1 Q 2 . Hence the ideally giealost output of work is i elated to Q 2 , the heat i ejected a I the lowci limit of tempeiatme, by the equation rvQA^-TJ T 2 ' A coiiespondmg pioposition m the thcoiy of icfugeiahon is lh.il the ideally gieatest coefficient of peiioimance of a icfiigcinhiig machine, woiking to pump up heat fiom T lo 2\, is obtained when the machine is theimodynamically ^eisiblc. In lh.il disc the same lelation holds, namel} and the amount of woik W Mhich is spent in duving Hie machine (and is> equal to Q x - Q,) is iclated to Q.> by the equation w _^( r l\-T 2 ] ~ T * 2 In othei woids, the gieatest amount of woik that is theoretically obtainable in letting heat pass clown through a qivon range df tempeiatme is the least amount of woik thai will suffice lo pump up the same quantity of heat thiough the same range To show that no lefngeiatmg machine can be more cfucienl than one that is reveisible, we shall use an aigiimenl like thai of Ait W Let E, fig 34, be a leveisible lefngeiatmg machine, leverscd and IV THEORY OF REFRIGERATION 135 Q, = Q 5 R theiefoie serving as a heat-cngnie. It takes a quantity of heat, say Qi> fiom the hot body and delivers a quantity Qa to the cold body, converting the difference into work. Let all the work W which it develops be employed to drive a lefngeiatmg machine R, and assume that there is no loss of power in the connecting mech- anism Accordingly the two machines, thus coupled, fonn a self-acting combina- tion. If it weie conceivable that the machine R could have a gieatcr coefficient of pei loi mance than the ie\eisible machine E, that il0 would mean that the latio of Q 2 to II' would be gieatei in R than in E Hence (II' being the same lor both) It Mould take moie heat fiom the cold body than E gi\ es to it, and R Mould also gi\ e moic heat to the hot bodj than E takes fiom it The icsiilL Mould be a continuous tiansfci oi heat fiom the cold bod\ to the hot body by means ol j. puicly sdf-aclmg agency This would be conliaij to the Second Law ol Thumod\ nainics we conclude theieloie that no ic- t'rigeiiitmg maclunecan havea highei coclhcientol peifoimanccthan aicveisiblc machine woikmg between the sainclimilsoltempeiaturc. It follows that all ic\cibible icfiigeiating machines, MOikmg bclwcen tlie same limits of tcmpeiatuic, ha\c the same coefficient of peifoimance. It also follows that the value of this coefficient is thai which would be found in a icvcised Cainot cycle, namelj W = r J\ - T, ' This is the ideally highest coefficient it mcasuies the peifoimance of what may be called a pci feet icfiigci al ing machine The coefficient of peifoimance in any real machine is neccssaiily less, foi the cycle of a real machine fallt, short of icveisibihty 107 Conservation of Entropy in a Perfect Refrigerating Pro- cess We saw in Art. 45 that a pcifcct, or reveisiblc, heat-engine, 136 THERMODYNAMICS |rn. such as Carnot's, may be legaidcd as n, device which Inmsl'ci'- entropy from a hot body to a cold body wiLlioul, allerin^ UK amount of the entropy so transfcucd, allhou^h the ainounl of heat which enters the engine it, grcatci than the mnotiHl of Ju-nl which leaves the engine. The entropy to ken liom Uu; hoi body, namely QJTj^, is equal to the entiopy given lo I lie oold body, namely Q. 2 JT Z \ it may be said to pass Ihi'ough Iho engine wilhoiil change, though the heat that passes through is nduetd in I he piocess by the amount which is con veiled mlo woik, lunncly, by the amount Q L Q . Similaily a peifect, or reversible, rcfngei'alinn niachmo or lx.il- pump may be regaidecl as a device which hansleis enliopv hoiu a cold body to a hot body without alluing Uie amount <>| Uu entiopy so tiansfeued, although the amount oJ lu.il whu-li eiilt is the machine is less than the amount which leaves Ilio machine. The action is in eveiy particular a, icversal of lh.il of the ju idcl heat-engine. Entropy to the amount QJ f J\ is I. ikui fioin Ilu cold body, and entiopy to the equal amounl <y,/7', is ^i\cii lo I he waimei body to which heat is disch.nned The .iinoiinl of lu.il wbch is pumped up mcieascs ftom Q, to Q { in I IK piocc-ss, because an amount of woik equivalent to Q L Q, is cvpcndcd in diainy the machine and is conveited into heal williin I In niachiiK 108 Ideal Coefficients of Performance The follou HIM lalih shows the \alues of the cocflicicnL of pcifoiiuanec in tl pulid 01 leversibleiefrigeiatmopioccss, loi vanoiis lanyeM)! (cinpci.ihin These aie ideal Homes, leprcscnhno <L lli(. ,clu'.il liniil u|,i<|, cannot be i cached in piaclicc Thouoh I hey ulalc lo coiidilions of levcisibihty which aic nol fully allaiuabk- in a ical niacluiK , they illustiatecleaily lhcpiacliCiilini]K)i lance <l in,ikiii K Ilu i.mn/ of tempeiatuie as small as possible, by lakini> in UK IK at til a l< m- peiatuie no lower than can be helped and by disdi,irni ni , il nll< i the least piacticable use Coefficients of Pafoimancc of a Pojnl Ktjn^itilnm Mnclnn, 00 I I I I n n 7 J Lo\\u Ijinifc of teinpeitituie U|>])u limit, ol |,( l!ll|)( 1 l( [|,,\ (Centigiade) 10 20 J 30'" " (in; 40" - 20 U 8 I <> .{ fi 1 1 2 - 15 108 7i r>i 1 7 - 10 Li I So , . 3 t) r> '{ - 5 179 107 77 It 273 13 !) I (! N 5 55 b 18 fi [] l 7 (1 ) 10 28 3 14 1 !) 1 ivl THEORY OF REFRIGERATION 137 The importance of a nairow range of temperature in rcfrigciation is fulther illustrated by T F~ f 1^ fig. 35 It gives the en- ' ''< _ , , 00 A Lf.' WV,</*,M < i &/,*,/,, ' ' \ (m tropy-tempeiature dia- gj ^ ^ grams of tliiee reversible -3 B **^ ^ -^ T z (b) i efngeratmg processes, m 03 \ all of which the upper g" Q ^/^.^ T 2 (c) limit of tempciatuie (2\) ^ is the same, and the same amount of v^ork is spent. Each of the three Entiopy D supposed piocesses is F J g ^ ideally efficient il is a icveiscd CarnoL cycle, and its cntiopy- tempeiatuie diaijiam is a icctangle. The aiea ol the lectangle repicsenls the woik spent, and the aica under it, down to the absolute zcio of tempciatuie, icpicsents the amount of heat that ib Liken fiom the cold body, and llieitl'oic mcasuies the leingeiat- mg clfccl The tlucc pioccsses foi \\lnch the diagiam is sketched dillei only in the temperatmc T 2 oi the cold body fiom which heal is cxhacted Thai Lcmpciahuc is iclativch high in the inst case (a), lowci in case (b) and ]oA\ei still in case (t) The iciiigciahng cflccl is measuied by the aiea^li) in the fust case, by CD in the second, and In CD in the thud The lesull of lo\\eiino T 2 is ACI\ appaient, in leducing the amount of icfiigciation thai is ulcalh capal>lc of being done In a gi\ en cxpcndilnic of \\oik 109 The Working Fluid m a Refrigerating Process The \\oikmg subslancc in ,i ulii^eiating c\clc may be a yas \\hich icmains gaseous tlnoiiglioiil, such as an. Moie commonly it is a (hud which is ,iltcinalcl\ coiiflcnsed and c\apoialcd Dining cxapoialum <il a Jow picssuic the lluid takes in heal fiom the cold bod} it is then compicsscd and ones out heal m becoming con- densed at a iclativcly high picssuic The selcclmn of the fluid is govcined by piaelical consideiations AValci is used m some cases but a sciious diauback to its use is the vciy laigc \olume and low pressure of the vapoui at low lempeiaLmes. Thcie aie obvious advanlages m using a fluid whose vapoui -pressuie is neithci mcon- vcmcnlly small at the lo\\ ei limit of tempciature noi mcoin cmcntly large at the upper him I The fluids most commonly used are ammonia and cai borne acid. Ammonia has a veiy convenient lange of vapour-pressure throughout the lange of temperature with winch we are concerned in piactical refngeration, With caibomc 138 THERMODYNAMICS [cir. acid the vapour-prcssuie is considerably higher, the culical point is leached at a bempeiature that may come within I lie ninge of opeiation, and the theimodynamic cfllcicncy is somewhat less. Notwithstanding these objections caibomc acid is Jicqm-iilly pre- feired, especially on board ship, where it is moic hainik'ss slioultl any of the fluid escape by leakage into Ihc room ecu laming I ho machine. Foi use on land, especially whcie I ho highest Iheimo- dynamic efficiency is aimed at, ammonia is usually chosen Olhei fluids with lowei vapom -presumes arc occasionally used, such as sulphurous acid, ethyl chlonde, and methyl chloride. no. The Actual Cycle of a Vapour-Compression Refri- gerating Machine. If the icversed Cainot cycle were acliuilly followed, the choice of woikmg fluid would make no dilfeienco lo the efficiency the coefficient of perfoimance for any lliud would have the value shcmn in Ait 106, namely T,/('J\ - T,} Bui a part of the icveised Carnot cycle is omitted in practice, wilh (he lesult that the coefficient is i educed, and thecxlenl ol Ihe mlucliou depends on the natuie of the fluid, it is grcalci 111 caihomc acid than in ammonia To cariy out a reveised Carnot cycle, wilh srpauilc- organs foi the successive events which make up the cycle, would icqum- (1) A compiession cvhndei in which the vapoiu is compu ssc d fiomthepiessuiecoirespondmg to T, to IhcpicssmeootrcsMondiix. to T! " ^ (2) A condenser in which it is condensed a I 'J\ A typical form of this organ would be a sinfacc condcusci in ulnch [lie \\mkiM fluid gives up its heat to circulating watci (3) An expansion cylmdci in wluch it expands fiom V, lo 7',. (4) An evaporatoi in which il takes up heal ,il T, IK.III tlu- cold body fiom which heat is to be cxtiacled This x c ssr! Is some - times called the "refngeiatoi." In neaily all refiigeiatmg machines the expansion cyl.ndn is omitted foi icasons of piactical convenience, and I lie lluid sln-ains fiom (2) to (4) through a thiottle-valve with an ad,uslal>l( opening, called the ' legulaLoi" 01 "expansion-valve" In pass,,,., ll.o c-x' pansion-valve the pies.uie of the woikmg fluid falls lo IhM of Ihc evapoiatoi its tempeiatuie falls lo T, and pad of ,l | immit . s evapoiated beloi'e it begins to take in heat fiom UK- c-old body The omission of an expansion cyhndc,, w.lh the snbslilulion lo, it of an expansion-valve, lc duces the cocflicicnl of peifonaanoc for IV] THEORY OF REFRIGERATION 139 two reasons The woik which would be reco% eiecl in the expansion cylinder is, lost, and also the lefngerating effect in the evapoiator is reduced, for nioie of the liquid is vaporized in the act of sti earning through the expansion- valve than would be vaporized in achabatic expansion, consequently less is left to be evapoiated by subse- quently taking m heat fiom the cold body. The loss of efficiency from these U\o causes is not, however, very impoitant undei ordinal y conditions To omit the expansion cylinder is a consider- able simplification of the machine, all the moie as the effective volume of such a cylinder Avould need adjustment i datively to that of the compression cylmdei in ordci to sccuie the best effect undei vaiymg conditions as to the limits of tempeiatme Rather than Fitf JO Ui<riin- ot a V ijiom ( otiipi< s-mn Ki li in( ml m<j, M.uluiH inLioducc llus complication it is woilh while io make a slight sacuhce ol Ihcimodynannu ciricKiicy In the usual type of vapoiu-o'oinpixssion icli isolating machine, aecoidmglv, the expansion cylinder is onulled, and Hit oigans are those shown diagrammalically in fig 3G They aie, (1) Die com- pression cylinder /?, (2) a condensci A such as a coil of pipe, cooled by ciiculalmg walcr, in which I he working substance is condensed undei a iclalively high ])iessiue and al the upj)c i i linul of tcm- pciatuie T I} (J3) an cvpansion-valvc 01 icgnlaLoi 7? Unough which it sti earns fiom A to C, ( 1) the evapoiatoi C\ in which it is vapoiiml at a low picssuie by taking in heat fiom the cold body at the lowci limit of temperature The e\ aporatoi may foi instance be a coil oi pipe taking in heat from the bin rounding atmosphcic of a cold chamber; often it is a coil suiiounded by cold circulating brine 140 THERMODYNAMICS \ni. which serves as a vehicle for conveying heat Lo the woiking sub- stance liom a cold chambei 01 fiom cans for ice-making 01 olhoi objects that aie to be lefngerated The action of the compiession cjdmder is shown by the indicator diagram, fig 37, m the same figuie. During the ibrvvaid stroke of thecompiessoi the valve leading to A is shut and that leading from C is open. A volume V^ of the woiking vapoiu is taken in fiom C at a umfoim piessme coiiesponding to the lower limit 2' 2 . In most actual cases what is taken in is not diy-sal mated vapour but a wet mixtuie, the wetness of which is regulated by adjusting the expansion valve R. This is in ordei that the subsequcii t conipi cssioii may not produce much (if any) supeiheatmg It is possible lo make the compression wholly "wet" by taking in a sufficiently wel mixtuie moie geneially the expansion-valve is adjusted so th.it the vapom is modeiately wet to begin with, and becomes s \/ JFig 37 Indicatoi Diagiam of Compiession Cylmclpi supeiheated by compiession. At the end of the I'oiwaid sliokc UK \alve leading fiom C closes and the piston is foiced to mo\c b^ck compressing the vapom 01 wet mixtuie in Ihe e\linclei until Ms piessuie becomes equal to that in A Tins compicssion i educes the volume of the fluid in the cylinder to V^ The valve lejidmg lo A then opens, and the back-stioke is complclcd nuclei* a nniloiin piessme uhile the woiking substance is dischaiged mlo A <iml condensed theie The valves of the compicssor aic spimg v.ihis wliich open and close automatically in consequence of Llie ch;ingc-s in piessme, and are situated in the covci of the c^lmdci in suc-li a mannei as to make the clearance negligibly sm.ill To com])U It- the cycle, the same quantity of woiking substance is allowed lo pass dnectly from A to C thiough the expansion-valve A'. This step is not leveisible (Ait. 22) The tempeiature T l at which condensation takes place, is in piactice necessarily a good deal higher than that of the eiiculalmn watei by which the condense! is kept cool, for a laige arnouul of IV] THEORY OF REFRIGERATION 141 heat has to be dischaiged from the condensing vapour in a limited time. But it is impoitant that the condensed liquid should be no warmer than is unavoidable bel'oie it passes the expansion-valve. Accordingly the condenser is ai ranged (sometimes by the addition of a separate vessel called a "cooler") so that the condensed liquid is brought as neaily as possible to the lowest tempera tme of the available watei -supply before it passes the valve, though it may have been condensed at a considerably higher tempeiature. The b d g f (( h Pig 38 Tho Vdpoui Completion Cyclo, u-inn Ammonia I Fiy JO Tbo Vapoiii Compicssion Cyolo, using Carbonic Acid advantage of I his will be obvious when we considei, in the next u tide, the theimal cflects of each step in the cycle in. Entropy-Temperature Diagram for the Vapour-Com- pression Cycle. The complete cycle is exhibited m the entiopy- bemperature diagiam of fig. 38, which is diawn for ammonia as \voiking substance, and fig 39, which is drawn foi carbonic acid Tlieie dg and ch are portions of the boundary cuives. The point a represents the condition of the mixture which is drawn into the 2ompression cylinder, when compression is about to begin; its 142 THERMODYNAMICS [ CI1 - wetness is measured by the ratio ah/gh. The line db lepiesents adiabatic compression to the pressure of the condenser. The next piocess consrsts of cooling and condensation at this constant pres- sure it is made up of thiee stages, be, cd and de In the fiist stage, be, the superheated vapom is cooled to the tempeiature at which condensation begins, in the next stage, cd, the vapour is completely condensed, in the third stage, de, the condensed liquid is cooled to the lowest available tempeiatuie before it passes the expansion- valve. The lines be, cd, and de form paits of one line of constant pressure In fig 38 de is practical^ indistinguishable fiom the boundary line, but in fig. 39 the distinction is veiy appaient be- cause we aie there dealing with a liquid that is highly com- ^ pressible in consequence of its el \ nearness to the critical state. A \ The line efrepiesents the pro- ' \ cess of passing thiough the f expansion valve, m which the piessuie falls to that of the evaporator. This is a throttling piocess, for ^hich/ is constant (Ait. 72) ef is theiefoie a line of constant total heat, its direction changes m fig. 39 m crossing the boundary cui\e. Bypassing the expansion- vah e the wo] king substance comes into the condition shoun by the point / The proportion which is converted mlo vapour by the meie act of passing the valve is sho\Mi b}' the ratro gf/gh. Lastly we have the process of cflcctn c evaporation when the substance is uscfidly extracting heat from the brine or other cold body by evapo- rating m the refrigerator This p 40 n is represented by the line /, during which the d^ness changes from gfjgh to ga/gh iv] THEORY OF REFRIGERATION 143 The refrigerating effect, that is to say, the amount of heat taken in from the cold body, is represented by the area under the line fa, measured down to a base-line corresponding to the absolute zero of temperature, namely the area mfan (fig. 40) The amount of heat i ejected during cooling and condensation of the vapoui and subsequent cooling of the condensed liquid, is the aiea under the lines be, cd and de, namely the aiea nbccleo The thermal equivalent of the woik spent in canymg the woikmg substance thiough the complete cycle which is simply the work spent on ib in the compiessoi is the diffeience between those two quantities, namely the aiea nbcdeo minus the area mfan. It should be noted that the work spent is not measuied by the aiea abcdefa, Fi# 41 < yU<> fin (Jaihoiuc Acid, \\ilh compiLision above the (Jnfica.1 Pressiue enclosed bv I he lines which icpicscnt the complete c\cle, because the cycle includes an mcveisible slop ef (sec Ait 51) In consequence ol that the woik spent is giealei than the enclosed aiea by the amount ocfni As a fuilhei example we may take a compression piocess (fig. -41), with cai bomc acid foi woikmg substance, m which the tempeiature of the cooling water is so high that the piessure dining cooling is above the critical piessuie The line be is accoidmgly a continuous curve lying entirely outside of the boundary curve. The woikmg substance passes from the state of a supeiheated vapour at b to the state at e without any stage coirespondmg to cd in fig 39, m which it is a mixture of liquid and vapour As befoie, the refn- g-eiatmg effect is measured by the area under fa. the heat rejected to the cooling water is measured by the aiea under be the difference THERMODYNAMICS [en. between these two quantities measuies the woilc spent, and is greater than the area of the closed figui e ale/a by the area undci the me \eisible step ef. 112. Refrigerating Effect and Work of Compression ex- pressed in Terms of the Total Heat While it is ins Line Live to state, as in the pieceding article, the lefiigcratmg eflccl, the work of compiession, and the heat lejecLccl, in teims of areas on the entropy-lempeiatuie diagiam, it ib much moie useful, i'oi purposes ot piactical calculation, to expiess these as follows m teims of the total heat of the substance at the vanous slaves of the opeiation. The lefngeratmg effect, that is to say the amount of heal laken in fiom the cold bod)'-, is / - I f , wheie I a is the total hciL a. I a and Ij is the total heat at/. This is because the (icveisiblc) opera- tion fa is effected at constant piessuie (Ait 68) Foi Ihc s.ime leason the amount of heat lejected to the condensei and coolt'i is I b I L , wheie those quantities designate the total heat t j L b and at e respectively Fmthei, in the piocess cj of passing Llio expansion-valve theie is no change of total heal, by the pimciple pioved in Ait 72. Consequently, I f = I e . We may Ihcielou 1 stale the amount of heat reacted as I b I, Again, the work spent in the compiessoi is (in thcimal units) I b I a . It is the theimal equivalent of the aiea of the indicator b diagiam in fig 37, namely A VdP, which is equal to /,, 7 ft |>y a the geneial principle pioved in Ait 80. We are dealing licie wil h the mciease of total heat in adiabatic compression instead ol ils deciease in adiabatic expansion That these lesults are m agi cement with one another is seen by considenng the heat-account ot the cycle as a whole Woik spent = Heat icjected Heal Liken in 1,,-Ia = (I* -If) ~ (la -I,) The coefficient of peifmmance, winch is the ratio ol the heat Uiken in from the cold body to the work spent m the compressoj , is la -If I*-Ia It will be obvious that the numerical value of this coefficient would be icduced if we weie to omit the cooling after condensation, which is repiesented by the line cle For in that case / would be iv] THEORY OF REFRIGERATION 145 shifted Lo flic light, Lo a point on a line of constant total heat through d, and I t would be mci cased. The refrigerating effect would be lessened, but the woik spent in piodiicmg it would be the same as before, foi the indicator diagram of the compiession piocess, which is measured by / - J a , is not affected. The values of I a and l b depend only on the state of Lhe substance at a and at b icspcctively, and aic the same as before. 113 Charts of Total Heat and Entropy for Substances used in the Vapour-Compression Process. The above results will show that calculations of peifoimance, as regaids refiigeratmg effect, heat icjected, and woik expended, become very easy when we can find the total heat of tho liquid just befoie the expansion valve and that of the vapour befoie and aftci compression This is leadily done if data aic available for chawing a Molhei chart of cntiopy and total heat loi the woiking substance Fanly complete elata arc available foi ammonia, cai bonic acid, and sulphuious acid Iff) charts foi these substances Mill be found in a Repoit of the Refngeialion Rcseaich Committee ol the Institution of Mechanical Engmecis ' . In dialing these chaits a gecmictiioal device is lesoitcel to for tlic [)urj)osc ol making the cliagiams at once open and compact, Milh tlie cflect that measuiemeiils ma} be made \\ith sullicient accmacv on a ehait of icasonable si/e This ele\ ice, which Molhei 01 igmallv adopted in chauino lus 7r/> ehail toi cat borne acid, is to use. oblique cooiehnates, as illuslialcd in lig J2 The lines ot con- st, ml / aie hon/ouLil the lines ol constant </ nisteael of being pcipenelieulai to them aie inHiiud il a small angk The icsiilt is that ulien the chail is di.iun the eui\es on it aic shcaicd o\ci, .is compaicd with the I'oim lhc'\ Mould lake on a chait \\ith iccl- angular a\( s \\\v figuie \\liicli M r hcn diaun \\ith lectnngulai cooHluiales is i(lati\el\ Ie>u<> in e>uc diagonal elneelion may Mifh advantage be 1 opened out by the ILSC ol oblique cooidmales This * J\lui 1'iDi In^l Muh I'JiK/ , Uct I'lll Tlu i luu ts gi\ on Lhcu iiu chawn by Piolos^oi. ( ' K ,J( nkin Tho (halt foi (.uljunic IK id nnlinclic's icsulLs ot oxpcinnents hy MISSIH ilciikiii ,iiid Pj'u on tlio Llii'imal piopcilios ol thai suhNtaiico (Phil Tinn^ Roil tiu< , vnl i \ I'M), p (>7 and A 53J p 35}, which involve Homo collec- tion of an otulic i t liaii ])ul)Iiilu'd h\ r JJi IVrolhoi TIio data foi aintnonia arc those ^ivoti by Alossirt (foocleiioiiyh and Moshoi (Bulletin No <>(> of tho University of llliiifiiH, I'Mtt) IMoiu iCLontly, conii)lolo liibloa oi Iho ihouiiodynanuc piopoitios of ammonia have boon ciitculalod with some\vhat diitoicnt numencal results by MoHSisKoyos and Riowtiloo (New Voik, John Wilov and Sona, 1910) In each of those publications a Molhtn /</i chaib is included u T 10 146 THERMODYNAMICS C'll is tme of I(f> charts, as applied to them, the deuce gives a beltei separation of lines that run moie or less diagonally acioss the sheet, like the lines in fig 31 (Ait. 101). Thcie is consequently a gu-al gam in clearness and in ihe power of accurately measmino I host changes of I that take place in lefngciatmg piocessos The in- clination selected foi the oblique axis will depend on I lie deyiee of opening out that is convenient in any particulai chart. In the case of fig. 42 it is 5 along the slope to 1 veilically, uid hence a measurement of I if made along a line of constant </> would have lo 1 = Fjg 42 Use of obliquo coouhuilm in llio l</> < liul be interpreted on a scale five times as CCMISC- as UK nciniiil sc-.ilc- ioi /. An 1$ chart foi ammonia diawn wilh ohlujiu coouluLilcs is shown (on a small scale") in fig. d3 In llus c-asc llu- amounl ,,r sheaimg is moderate, foi the slope ot Ihe lines o! consla.U onliopy is only two to one. The dmgumi, foi Ihe useful .(<.,, ( , )n s,sls c,f a fan-like gioiip of hnes of constant pressure cvUii(lin M as si uuliL hnes though the legion oi wetness JVom Uu- JU,MK! I.(,.,,HL,,\M(. the vapour bounclaiy or saturation curve, and HUM, ,, s ( m ^ os mlo * For snrnlar charts i a m lulloi detail ami ou a male, I,,,,,,,, n <,wh f,u UH. ,u lems, reference should bo mado to tlu, IV] THEORY OF REFRIGERATION 147 the legion of superheat. Lines of constant temperatmc aie also drawn in the region of superheat, and lines of constant diyness (shown as broken lines in the chart) are diawn by dividing the 43 /</) charL foi Ammonia sLi'dight poiLion of cacJi line of constant pressnie into a number of equal parLs Tins chai t should be compaied Avith that shoAvn for \\atcr and steam in fig 31 (Art. 101) m which, ho \vevei, there was no shearing, for lectangular cooichnates weie employed. Allowing 102 148 THERMODYNAMICS [CM, i'oi that difference the leniarks made in Ail 101 apply lieu 1 . The slope of any constanl-pies.suic line, when piopcrly mteiprcled with reference to the coordinates used in (he. drawing, nieasuves the tempeiatuie, fbi T dljch/^ Ilcnc'e iheie is no abiupl cluium 1 of dnection between tlic stiaii>h( pail of ciny such Jinc and ils / 70 41 !(/> cliait, fui (Jiulinnu An curved conciliations mlo Lhc liquid legion a I one end and into the legion ol'supeihcat al Uie olhei This ol eouise applies lo any substance. The /</., ehaiL foi siilj)luuoiis acid j,s ^eneially sinulaV to the chait for ammonia The Icj) chart foi caibonic acid is shown on a small s( ale in fi<> J, I-. It showt, the icj-ion ]ound abouL I he ciilienl point. Thai point IV] THEORY OF REFRIGERATION 149 coincides with the point of inflection of the continuous boundary curve ( Ai 1. 101 ). Constant-pi essure lines are drawn foi pi essures that are higher than the cutical pi essure as well as foi the wet region. The principle already stated applies to these lines, that the slope at any point (due regard being had to shearing) measures the tem- perature. In passing up along any line of constant pressure above the critical pi essure, the slope, which mcasui cs the tem- per atuie, mci cases continu- ously 1 . The stiaight poitions of the constant-pressure lines, within the boundary curve, aie divided by biokcn lines which aie lines of constant diyness Lines of constant Icmpeiature aic also diawn in the legion outside the boundary ctu\ e In thcicgion \\rlhm the boundary, where the state is that of a mixture of sal mated \apoui and liquid, these lines would of comsc be straight, and would coincide with lines of constant pic'ssiuc. To a\ oid confusion the sliaiyht portions ot the constaiit-tempcialurc linos aie omitted in the IIIHUC 114 Applications of the I(f) Chart in studying the Va- pour-Compression Process) We aic now in a position to represent the vapour-comprcs- H! As Moaais .Jonkin and Pye h.uo pointed out (lou at , p 305) in eoirecting the u.ulmr c luut of MolJior, them is no point of inflection in any ol these lines For, since ( (IT \ , winch is a positnc quantity throughout ihe whole r 45 Rofijgciation e\cle t-aced on. the /(/> chait foi Cailiunic Acid dl\ =T d<l>Ji> ' COUIHO of any lino above the cubical piosstno, as will he seen hy reference to the onliopy-temperatiuechagiam A point of inflection would icquire (^2) , to be zel Siuno of Lho constant jiiossuio lines \\cre ononeously di&vm \\ith inflections in Mollici's ougmal lef> cha?t foi caibonic acid, winch was lepioduced m the author's book on The Merhaniuil PindiiLtion of Cold | Parts ot this HI tide aio L,ikcn fiom an appendix (by the pieseut wiitei), to the Ropoifc of the Rofijgciation Kcacarch Committee of the Institution of Mechanical Engineers, 1914 15 o THERMODYNAMICS [cir. sion refrigerating process by diagiams which exhibit the changes of total heat in relation to entiopy. With the help of Itf) charts numerical values of the total heat are readily found by measuic- ment at each stage m the assumed cycle. To tiace a lefngerating cycle on the appropriate chart, begin as before at a point a (fig. 45) which icprescnts the state of the sub- stance when it is about to enter the compiessor. This point is on the constant-piessure line corresponding to the process of evapora- tion in the cold body or evapoiatoi (fig. 36), and ils distance from the tovo boundaiy cuives coiresponds to the propoition of vapour to liquid in the mixtuie. If the compression is to be completely "diy," a will be on the boundaiy curve (at a^) more gencully tlic substance is slightly wet when compression begins. The stuight line (tb, diawn paiallel to the lines of constant entropy on the chart, is the piocess of adiabatic compiession The position of b is detci- mmedbythe mtei section of this line with a Jme ot constant prcssmc corresponding to the known upper limit of pressuie at which con- densation is to occur The temperature reached in the piocess of compiession is seen by the position of b among the lines of cons! an I tempeiatuie. In geneial there will be some superheating Bui il the mixtuie is so wet to begin with that the adiabatic line Ihiough a does not cioss the boundary cuive dining compiession bdoie the uppei limit of piessme is icached theie is none, and in Hint case the piocess is spoken of as " wet" eompiession. This would lie the case if compiession had begun at a c instead of a. J$y beginning at a it carries the substance into the legion of snpcihcat belorc compression is completed at b Next we have the constant-pi CSMH c process of cooling and condensation and fmlher cooling, icpic- sented in its thiee stages by the lines be, cd, and dc, the posilion of e being fixed by the tempcratme to which the liquid is known to be cooled befoie it reaches the expansion-valve. Then a hoi i- zontal stiaight line thiongh e (a line of constant total huil) repiesents the piocess of passing tlnough the cxpansion-vnh c , and determines a point/, on the evapoiation line, which exhibits the condition m which the substance enLcis the cvnpoiuloi. TIu process of evaporation Ja, which is the effective icfiigcinlmg process, completes the cycle The values of I a , 7 & , I c and I L (which is the same as I e ) aie lead directly by measurement from the chail As has been already pointed out, the woik spent in compiessmi> the substance is I b - I as and the icfugcrating effect is I a - I f We may illustrate the use of the chart by some examples. Take first a case in which the working substance is carbonic acid, with iv] THEORY OF REFRIGERATION 151 10 C. as the temperalme of evaporation, 25 C. as the tem- peratme of condensation, and 15 C. as the tempera tuie to which the substance is cooled before passing the expansion-valve. The diagram for the perfoimance of an ideal machine undei these con- ditions is sketched m fig. 45, assuming various degiees oi dryness at the beginning of the compiession. If the substance is then entaely dry the operation staits at a lt namely, the end of the evaporation line for 10 C., and compiession brings it to b which is on a line of constant piessme equal to the piessme of satuialcd vapour at 25 C , namely, 930 pounds per sq inch. But the vapour is considciably supei heated at b lt its temperature there (as the lines of constant tempeiatuie show) being 58 C The woik spent in compression, Avhich is most accurately found by leading off the length of the line a^b-^ on a scale A\hich makes that length a dnect mcnsiue of the change of/, is 8 7 We next tiace the piocess of condensing and cooling, under the constant picssmc of the condensei. Fiom b 1 to c the gas is losing ils supeihcnt, fiom c to d il is being condensed, and fiom d to e il is being cooled as a liquid. The point e is found b\ the intellection oi the line of constant piessme undci \\hich the piocess is cauicd out with the line of constant tcmpciature foi 15 C Nevt diaw f/paiallcl to the lines of constant total heat to meet thce\ apoialion line loi - 10 C The refiigualing effect / (li - /, is -179 The cocllicicnl of pcilbimance is theicfoie 5 5 This cycle con expends to completely diy compiession Suppose on the oLhci hand thai the compiession is just v\ct enough to dvoicl an\ supcihealmo In lhat case il musl commence lit a c in oulei Ihal llic adiabalic hue lepu'scntmg the compiession ma} pass lluough c on the hoimelaiN cm\e Then Ihe woik done in compiession is smallei than beioie 1 , foi a,c is smallei than a^b^, The icfngeiating efle-el is also smallei, foi fa, is smallei than Ja^. The c-oc'dicienl of peifoimance is now found to be 5 51- Bel ween these two thcie is a ceitain dcgice of diyncss which gives a slightly lughci cocfTicicnt of peifoimance than eithci This may be shown by laking a succession of poinls foi vanous stales of diyncss between a c and a as the staitmg point of the cj cle, and woilong out Ihc coefficient of peifoimance foi each But we may leach the same conclusion moic duectly as follows, by a general melhod which is applicable to any I(j> chait The icfrigcratmg effect foi anv slate of initial diyness, a, is proportional (on some scale) to the length ja. The \\ork done is 152 THERMODYNAMICS |cn. proportional (on anothei t,cale) to the leiu-th ah Hence the position of b which Avill give the highest coefficient ol' peifoimance is that which gives the smallest ratio of ab to fa This is found by diiwmfc a tangent from/to the line of constant picssnie ou winch b lus J?y applying this method the point b has been delcimmcd in the iigme, and hence the point a ib found at which compiession should bei>in if the coefficient of pcifoimancc is to have Us maximum value In the example that value is 5 72, and is obtained when llu 1 imhul diyness is about 87. As anothei example, still with caibonic acid, I tike I lie same con- ditions as befoie, except that the condensed liquid, instead of beino cooled at 15 C befoie expansion, i caches tin valve at I he lein- peiatuie of condensation, namely 25 C. In thai case Ihe ])roe(ss of expansion conesponds to Lhe line rlf,[ in fin. 15, [he lesl of I he cycle lemammg as befoie For maximum coclhcicnl ol peilormancc, unclei these conditions, compicssion should no lon^ei slail 1'ioiu a but fiom a point so chosen that Lhe ad in bade line llnough il leaches the constant-pi essuie cuivc b^c at the ponil wlieie the lani>c nl lioin J a meets that cuive This coiicsponds to an initial <Ii vness ol about 95, and the maximum cocllicic.nl so obtained is [> ,'}<). When Ibis \ aluc is compared with that found in the pievious example, namely 5 72, it \vill be obMOiis that a seiions loss ol cllicuiicv is eausul b\ omitting to cool the condensed liquid betoie it i caches Ihei \pansion- vah e 1 " A tuithei example "Kill seive to illnstiate I be application ol llu 70 chait to caibonic acid \\oikmi> undei tiojiieal conditions, so that thelnghei limit of picssuic is alun e Ihe entical piessmeol llu subbtance Still taking- 10 (_' as Ihe teinpeuilnrc of <. vnpoi, >lion, we shall suppose the piessme in the condenser to lie 1200 pounds pei sq inch, and the tempeiahuc to winch llu liquid is eoolid befoie expansion to be 30 C 1 With these dala llu* diagiam I, ikes the foim shown in fig 1G, \\heic a^b^ lepiesenls a ])ioetss ol com- pletely diy completion, and ab a ])iocess of compression in \\hieh the position ot a has been <-o chosen as to i>ivt tlu- ma\imum co- efficient of peifoimancc The line ab consequently cuts Ihe euive of constant pi essuie foi 1200 pounds pei sq meh al I he place when a tangent fiom/ 1 would meet that euive. The point, r is delei mined *The nuuibeis givui in thosn ox.unplt^ A\CM^ louiiil l.y MKMIHIIICIIICHI, fioni ' . Committee of the Institution otMK]i.mit,ill':n K iii( CMS wi'[iMis.ilinHt( ml llu-iiiiiiilifiH would be slightly diftcicnf but the gcnu.il u'sults would not l.o ufk-i (ml [V] THEORY OF REFRIGERATION 153 by following the curve of constant piessure till it cuts the line of temperatme foi 30 C The maximum coefficient of peifoiman.ee is obtained \b, when the diyness before compicssion is about 95. Its value is 3 1, and undei these conditions the va- pom is superheated to 70 C al the end of com- piession The coefficient calculated foi completely dry compiession, when the compression line is fl 1 6 1 , has almost I he same value. In all these examples it is interest mo, anel practi- cally nnpoitant, to notice how little the coe lYicicnt ol pcifomuncc in the theo- ictical cycle is alfcctcd c\ en by considerable changes in the di \ncss be hue compicssion This is Line not onl\ of car- bonic acid but of any uoikmi> substance The application of the Mollicn tha^iam lo ammonia is illushated in lid -IJ, by an e\ani|)le iclciuny lo tiopieal conditions Theie, as in lonnei examples, UK lenipeiatiuc ol c\apoialion is taken as 10 C 1 The substance is supposed to be condensed at 35 (' (piessmc 107-3 pounds pt'i stj inch) and to icmam at that tcm- peialuie' until it leaches the expansion-valve The cycle is abcdja Foi ammonia undei these conditions the follow mo icsults aie obtained by measuiement fiom the chayiam foi vauous A allies of the initial diyness Ihvnoss TtofiiKciatinff Woik of (Jompu'Hsioa Fi 4(i Ri hini'inliOM cyili \\ ilh Cuboni' Ai nl win n the uppi i limit (it pi i ">MIK 1 i \u odi the LI jlii <i! [in SMiie Ttof updating Edoc'L Compii'ssiou I !)5 85 08 201 2 Hi 2.J04 2149 199 3 54 5 WO 454 425 398 CoolThrient of Peifoimance 480 502 507 50(i r )01 15 4 THERMODYNAMICS [LU. Here the maximum coefficient of pcifoimance is reached with a value of the initial dryness only veiy slightly gieatei than IhaL which just gives wet compression We may take the coefficient got by using wet compiession, with b on the boundaiy omvc, as piactically equal to that maximum. Fig 47 Rehigeiiitiou cycle with Ammonia uncloi Tiopiuil (JoudiUotiH As a final example, take ammonia wo i king in the same con- ditions as those that \\eie assumed foi carbonic acid in Ihc fust example, namely an evapoiation temperature of 10 C., a con- densation tcmpeiature of 25, and the liquid cooled lo 15 0. iv] THEORY OF REFRIGERATION 155 before passing the valve. We then have these results for various values of bhe initial dryness: D iv ness befoie Compression 10 Refrigerating Effect, 284 fi Work of Compression 422 Coefficient of Peiforraance 674 095 2091 384 701 09 2535 358 708 085 2379 336 708 08 222 3 315 7 Ob The examples agiee in showing that there is vciy little differ- ence in the thcrmodynamic efficiency of the ideal peifoimance whcthei wet compiession is used or the initial diyness is adjusted to make the coefficient a maximum. The mle for this adjustment, applicable to any woikmg substance, may be cxpiessed thus Maik a point fon the evapoiation line to show the state of the substance on entering the evaporator. This point is found by drawing a line of constant total heat fiom the point which icprcsents the state of the substance when it reaches the expansion-valve Fiom the point/draw /& tangent to the line of constant picssuic foi the condcnsei, touching that line at b. Then I he compiession line ab passing through b is the one \\hich gives the maximum coefficient of peifoimance in the ideal cvcle Avith adiabcUic compicssion It does not iollow that the same dcgice of initial \\ctness Mould give I he maximum coclTicicnt in a ical compiessoi, loi the per- loim<mcc of a real machine is complicated b^ tiansfeis ot heat between the working substance and the metal In general such li.msfcis will be less when the working substance is dry On the olhci h.md, with a wet mixUiic, what is called the volumetnc eflicic-Mcy of Hie apparatus is gieatcr, since a laigei quantity of I he working substance passes through the machine lor e\ery cubic fool swept through by the piston, and this tends to reduce the proportion of Ihose losses that arise from mechanical friction, and from radiation and conduction between the apparatus and rts environment 115 Vapour-Compression by means of a Jet. Water- Vapour Machine Whatever be the working substance, an essential feature of any vapoui-comprcssion refrigerating machine is that the vapour must be pumped up from the low-pressure legion in which it has 156 THERMODYNAMICS [cir. been evapoiated to the high-pressuie legion in which it is to be condensed. But this pumping up may be effected in move than one way The usual way is by means of a cylmdei and piston, and so long as the vapoui-piessme is modeiately high Lhe use of a compiessmg piston is quite satisfactoiy. But when the vapoiu- pressuie is veiv low, as it would be if water weie used fov ihc vvoikmg substance, the volume to be swept thiougli bv a com- pressing piston would be so laige as to be veiy inconvenient, and the amount of \\oik ^hich Mould be wasted thiougli fuel ion between the piston and cylmdei would be an excessive addition to the legitimate woik of compiession. Not only would Lhe machine be exceedingly bulkv but its practical efficienc}' -uould be exceed- ingly low At C , foi example, the density of waler-vapom is so small that about 3G5 cubic feet of it aie rcqiined Lo absoib as much latent heat as one cubic foot of ammonia vapour Hence to use watei-yapom as a lefngeiatmg agent some appliance must be icsoited to which will avoid the bulk and fnclional waste of an oidmaiy compiession pump One such appliance is a ccntiifuoal pump 01 icveised tin bine anothci is an ejector or jet pump, in which an auxihaiy stieam of vapom, supplied at a compaialivcly high piessme, Ibims a moti\c jet which drags with it the \.ipom to be "aspnated," namely the vapom which has been foinud bv evapoiation at low picssuie, so that both pass on tooolhci lo bi condensed This device is applicable to any ihucl, and closal- ciicmt sj'stems \\hich operate on this punciple ha\e lx.cn devised Foi othei woikmg substances besides u atci-\ apour The vajwiu of the motive jet necessauly mixes with I he \apoiu Lo Ix cispnalcd and both aie condensed togethei theie aie thus two cncuils \\hu-li coalesce in the condenser. PaiL of the condensed liquid reluius thiough the expansion-valve Lo the cold cvapoinLoi, and ,u-ls as the effective woikmg substance m producing icfimei.ilion llu other pait is foiced by a feed-pump into a boilci where iL is vnpo. .ml at a lelatively high piessuie, so that iL ma> ac-t as Ihc motive |d the t^o then meet again in the ejector on their way Lo (he c'-on- densei. It is however when watei is the woikmg substance Llul such a system is specially applicable An independent supply of |, O ,I CT steam foims the motive jet It acquncs a high veloc.ty , passmo thiough a discharge nozzle, which comcigcs lo a place al which the low-piessme vapom to be aspuated is allowed access The Inh velocity jet communicates pait of its momcntmn lo lhal vapour iv] THEORY OF REFRIGERATION 157 and the two pass on in one stieara to the condenser through a divergent pipe in which the stream loses veloci ty and gains pressure as it pioceeds. This enables the pressuie of the woiking substance to use from the lo\\er to the upper of the limits between which the machine works, namely fiom the low piessme at which the aspirated vapour is foimcd to the higher piessme at which it is condensed In lefrigciatmg machines constiucted to act in this way the quantity of vapour in the motive jet is as much as three 01 even five times I he quantity that is aspirated The thei mo- dynamic efficiency of the method is found on tual to be only moderate, but the apparatus has advantages m point of simplicity, and in the absence of any Avorking substance other than watei It has been applied not only to cool water, but also to maintain a tcmpeuituie considerably below C , in which case bime is substituted for fiesh watei as the woiking substance whose vapoui is aspuated, and the cooled bune is pi evented fiom becoming too dense by systematically letuimng to it a quantity ol watei to make good Ihe amount that is c\apoiatcd 116 The Step-down in Temperature Use of an Expansion Cylinder in Machines using Air So long as the woiking sub- stance in a relngeiating machine is a \apoui -\\hich becomes liquified dining Ihe opuation, it is piacticable, as \\e luuc seen, to dispense with an expansion cylmclci and still have a laigc amount ol iefii<>cialing elleet The slep-down in tempciatinc, uhich is nceessai} in any lefngciation cycle, occms as a, consequence of Ihe piocess of thioltlmg, while the subslancc passes the expansion- valve Tins is tun also of a gas neat its eiilical point, and hence a machine using eaiboiuc acid undci Uopical conditions can be ffectivc without an expansion c\lindci all hough the substance may not undeigo liquefaction A gas noai its culical point is \eiy lai fiom pci I eel and does not even appioximatcly confoim to Joule's Law A gas nluch conlonns to that law would suffei no -.Icp-down of lenipciaLuie in passing an cxpansion-vah e (Ait 19) With a gas such as an, which is neaily pcifcct at the tempeiatmcs ind prcssuies that occur in oidmaiy icfiigciation, the step-do vui uould be too small to seivc the clesned puiposc Hence with an P.OI woiking subslancc an expansion cyhndci becomes an essential ic'incnt of the machine Rcfiigeiatmg machines using air, and 20olmg it by means of expansion in a cylmdei in which it does work against a piston, are amongst the oldest effective means of 158 THERMODYNAMICS [en. pioducing cold by mechanical agency. They arc still used foi the diiect cooling of the atmosphcie of cold stoics, but their use is now less common, because machines in which the woikmg substance is a condensable vapom aie not only moie compact but give a better theimodynamic return for the woik spent in dining them. 117. Air-Machines. Joule's Air-Engme reversed. The au- machmes which aie in actual use opeiate by taking in a poition of air from the chamber that is to be kept cold, compiessmg it moie or less adiabatically with the result that its tempcialure uses considerably above that of the available water -suppby, then extracting heat fiom it in the compiessed state by means of cu- culating watei, then expanding it in a cylinder in which iL docs work, with the lesult that its initial piessure is icstoied and its Cuu'c.1 A Fig 48 Oigans of an Aii-Mdcliinc temperatuie falls gieatly below the initial temperatme. It is Lhcn leturned into the atmospheie of the cold chambci, with which it mixes, the object being eithei to lowei the tempciatuic in the chambei 01 to keep it fiom using thiough leakage ol heat iiom outride This type is known as the Bell-Coleman an -machine. The cycle is a icversal of that of Joule's Aii-Engme, descubcd in Ait. 5-1. As applied in lefngeiation the appaiatus takes the foim shown diagrammatically in fig. 4S In the phat>e of action shown theie the pistons are moving towaids the left Air fiom the cold chamber C is being diawn into the compiession cylmdei M. In the retuin stioke it will be compiessed fiom one atmosphcie to about fom, with the result that its temperature may be uuscd to 130 C. or higher. It is dehveicd under this piessuic to the tv] THEORY OF REFRIGERATION 159 coolei A wheic it gives up heat to the en dilating water and comes clown to near atmospheric tempera tin e It then passes, still at high pressure, to the expan- ,, sion cylinder N, wheie it does woik in expanding to the initial pressure of one atmo- spheic and theieby becomes very cold, i caching a tem- perature of peihaps 60 C. or 70 C , in which condition f it is ictmncd to the cold ehambei An ideal indicator v diagram for the whole cycle Fl S 49 Inihcaloi Dingiam of Aii Machine is gn en in fig 10, where fcbe shows the action of the compression cylinder and cadf shows that of the expansion cyhndei The aiea abed mcasuics the net amounted' work that is expended In the diagiam the compiession and expansion aic both tieated as adiabatic and the \ olume ol A as well as that of C is assumed to be so laige that dining delivery of Ihc an its prcssme does not sensibly change Willing T a , T,,, T L and T tl i'oi the tempeiatme of the vioiking an, at the points a, b, c and d of the chagiam, we ha\ c Q , = A p (T b -T n ) (bi the heat icjccted to the cooling watei, and Q, = K v ( r l\ - T d ) for the heat usclully extiactcd liom the cold ehambei The net Limounl ol \\oik expended is equal to Q.i Q.t The coefficient of peiloimanec is Q Q.-Q, Foi the icasou explained in Ait. 54' T T T T T T "=' a , fiom which ^' T ' l = ' J c l d * l> ~~ * a * b TT, ,, Q < ~ T " 1 q " T <- Jlcnce ,. - ,., , anci ,. r ., , . ' e i J b ( t i V< J- T, J- o This coelFieicnt of pcifoimance is low because of the M.IY laige iniige ol tcinpeiature Ihrough which the woiking air is earned For llns leason, and also because of gicatei fnotional losses, an Liclual an -machine gn r cs lesults that compaie unfavouiably with those obtained in the vaponr-compiession piocess. Consideicd as a means of pumping up heat fiom T c the tcm- pcratuie of the cold ehambei fiom which heat it, taken in, lo T a the tcinpciatine of the cnculatmg water to which heat is dischaiged the aii'-macliinc has two serious theimodynamic defects Theie is 160 THERMODYNAMICS |c. an llle^ eisible tiansfei of heat when the woiking air, aftei bung heated by compiession to 2\ comes into thermal contact vuth the cnculatmg watei at T a . and theic is another ineveisible tiansl'ei when the vsoikmg an, chilled by expansion to T lt , mixes with Lhe less cold almospheic oi' the chamber aL f l\ An ideally efficient icfiigeiatmg machine, namely a ie~\erscd Cainot engine, working between T u and T ( as uppei and lowci limits \\ould ha\ r c (Ail 105) a coefficient of pertoimance equal to The coefficient L'onnd abo\ e foi the ie\eiscd Joule cycle is sub- stantially less, because T b is highei than T a . In the piactical woiking of such machines the piesenee of moistme 111 the an has to be icckoned with The air coming from the cold chamber is more 01 less^saturated dining expansion it becomes super-satin ated and the water fiom it would IDC deposited as snow in the expansion cyhndei, and might mteifeie wilh Ihc action of the mechanism, it pievenLive de\ ices weie not mtio- dnccd One such device is to divide the whole expansion into U\o stages by making it compound In the fust stage the expansion is caiiiccl only far enough Lo cool the an to atemperatmc just above the freezing point In that way neail^ all the moist me is deposited in the foim ol watci, and is easil} diaincd away befoic the final stage, winch would ficeze it, begins. Anothei de^ ice is to condense out most of the moist me befoie expansion, b} passing the com- piessed an thiough pipes \\hich bimg its tempeiatuie down to near the fieezmg point befoie it enteis the expansion cylmdei. These '"cliyiiig pipes," aie kept cold by an iiom the cold chambei that an is consequently warmed by them, but the loss is made good by the lower lempciatme which the working an i caches in ex- pansion, as a consequence of the piecoolmg it has undergone in the drying pipes. 118 Direct Application of Heat to produce Cold. Absorption Machines In another class of icfiigeiatmg appliances theic is no application of mechanical powei the agent is heal, winch is supplied fiom a high-tcmpeiatuie source, and is employed in such a AS ay as to cause another quantity of heat to pass fiom a cold body and to be discharged at a tempeiatuie mtei mediate between that of the cold body and the hot souice In such machines the clli- ciency of the action fiom the theimoclynamic point of view is ] THEORY OF REFRIGERATION 161 easuied by the heat latio -~ where Q 2 is the heat extracted Ml! :>m the cold body, and Q is the high-tempeiature heat which is pphed to cany out Lhe opeiation. A typical example is the ammoma-absoiption refngeiating achnie Essentially this is a device in which the vapour of nmonia is allemalely dissolved by cold watei under a lelatively \v picssuie, and distilled fiom solution in water under a lelatively gh piessuie by the action of heat The ammonia vapour, driven f by applying heat to a solution is condensed m a vessel which kepi cool by means of en dilating watei. This gives anhydrous [incl ammonia at high piessure which (just as m a compiession achme) is allowed to pass thiough an expansion-valve, into a il or vessel foimmg the evaporatoi. A low prcssuie is maintained the evapoialoi by causing the evapoiatccl vapoui to pass into other vessel, called the absoiber, wheie it comes into contact Lh cold watei in winch it becomes dissolved. When the water the absoiber has taken up a sufficient piopoiLion of ammonia in tin n is heated to give of! Lhe vapoui agcTin unclci high piessuie the simplest foim of Lhe appaiatus the same vessel selves ematcly as absoibci and as genciatoi 01 distillci Foi con- uious woikmg thcie aie sepaiate vessels, and the iicli solution tiansfeiicd fiom the absoibei to the general 01 by a small pump, nlc the watci fiom which ammonia has been expelled (Ions back the absoibci to dissohe moie ammonia The scheme of such an pai a lus is shown m iig 50 Ilcal is applied to (lie solution in e ociicidtoi bv means ol a steam-coil. The <>as passes off at top Lhc coudcnsei, then thiouqh tlic e\pansion-\alve to the e^apol- >i, and I hen on to Lhc absoibci, wheie it meets a cmient of watu vciy weak solution that has come ovei fiom the bottom of Lhc ncniloi BelAvccn the gcncialoi and absoibci is the mtcichangei, levicc foi cconomi/mg heat by taking it from the watei that is Unnmg to the absoibei, and giving it to the rich solution that being pumped into the gcneiatoi This nch solution is dehvcicd Lhc top o( the column in the gcnciator, as the liquid paits \\ith e ammonia it becomes denser and falls to the bottom wheic it :apes to the absoiber thiough an adjustable valve When water soibs ammonia a laigc ninount of heat is given out Hence the sorbei as well as the condenser has to be kept cool by means of culatmg water or otherwise. Under the most favourable cou- pons the quantity of heat which such a machine takes in fiom 162 THERMODYNAMICS [en. the cold body is considerably le^s than tlie quanliLy ol' high Iciu- peratuie heat that has to be supplied, for it needs nioie theimal units to separate ammonia gas from solution in watei than simply to evaporate the same amount of liquid ammonia In another type of absoiption machine watei-vapoui is I he- sub- stance which is absoibcd it is taken up by sulphinic acid, fioni which it may again be sepaiated by the agency of heal Such a machine has been used for ice-making, the evaporation of part of the watei servino to fiecze the lest In this case also the heat Steam\ Retjulating Value Fig 50 Oigans of an Ammonia Absoiption M.uhiiR- ratio, namely the latio of heat usefully extiacted to heat supplied, is less than unity, for it takes moie heat to separate the vapom of watei fioni a sulphuric-acid solution than fiom pine walci It is a familiar fact that when watei is mixed with sulphuric acid much heat is given out It is obvious that a better thermodynamic result would be attainable if the piocess of absoiption of the vapour AVCIC attended by the giving out of less heat than is equivalent to the latent heat of the vapour itself. This is the case when ammonia vapoui unites with ceitain anhydrous salts, for which it has much affinity, such as the sulphocyamde of ammonium (NII 4 CNS), 01 the nitrate, biomide or iodide. Any one of these salts forms a suitable absoi- ] THEORY OF REFRIGERATION 163 nt. The ammonia vapour unites with the dry salt to form a [uid solution, fiora which the ammonia vapour can again be j ivcn oif by the application of heat, leaving the salt dry and i ady to serve again as the absorbent The vapour is stnctly ih vdi ous, lor no watei is present in the woikmg substance at any igc. The heat gi\ en out dining absorption of the ammonia j ipour by the salt is substantially less than the latent heat of e vapour itself at the same picssuie, foi pait is taken up in j] Kiefymg the salt. Similaily the heat icquned to effect ascpaiation ' ammonia vapour from the salt is substantially less than the j tent heat of the vapoui, foi part is supplied by the solidification the salt Consequently, when this piocess is made use of foi the , n pose of lefngeiation, the latio of the heat which is exti acted ' Din the cold body to the high-tempeiature heat, which is supplied ! Lhe gcnciatoi, would be gieatei than unity, if it weie not foi ch losses as occui through impeifcction in the working f This piocess is the subject of patents by Mi W W Seay*. In s icfngeratmg appaiatus the woikmg substance is made up of )out 8 Ib of the salt to one of anhydious ammonia. Theie aie io (or moie) similai vessels each of \\hich seivcs alternately as j )soibcr and as genciator These aie cylinder A\lnch aie kept )\vly icvolving as a means of slimng the nuvtuie Piecau- >ns have to be taken, b> selecting a suitable matenal ioi Lhe 'ssels 01 foi then lining, to avoid chemical action on the pait of ie salt In each vessel llieic is a eoil of pipe thiough \\Iuch cold -ilei ciiculatcs while I he \ essel is aclmy as absoibci, and hot watei steam while it is acting as gcneialoi The othei oigans aie Lhe me as in anv olhei compulsion 01 absoiption plant The nnionid \apoui passes J'lum I he gencialoi to a suilace condense i lieicits lalcnt heat is dischaiged to circulating watci, Ihentlnough i expansion-valve to Lhe e\apora,toi, where it takes up heat fiom e bime 01 othei body that is to be coolcel, and then passes on Lhe absoibei In the geneialoi anel condenser its picssuie is latively high in the cvapoialor and absorbei it is low Tesls of a heay machine show that, e\cn in small sizes, theie is much greatci amount of lefiigeiatmg effect foi the same expen- tuie of heat than is found in machines which work by the absoip- ni of ammonia in watci As applied to ice-making it appeals that e Seay machine will produce as much ice, per Ib of coal consumed, * BuUsli Patent (Marks), JSTo 25806 of 1907. 112 164 THERMODYNAMICS L cn - as can be obtained by employing a good steam-engine to duve a good vapoui-compiession lefngeiatmg machine. 119. Limit of Efficiency in the Use of High-temperature Heat to Produce Cold Any appliance, such as an absorption machine, foi the chiect production of cold by the agcncv of heat, icqunes a supply of heat at a tempeiatuie highei than that of the suiioundmgs. There aie nccessanlv tlncc tcmpeiatiucs lo be consideied. (1) the low tempeiatuie T 2 of the cold body fiom which heat is being extiacted, (2) the mteimediatc tempeiabuie I\ of Lhe available condensing watei 01 othei "sink" into which heat can be i ejected, and (3) the high tempeiatuie T of the somce from which heat is supplied to perfoim the operation Any such ap- Moto -j- ^i and ^ - Q | Refrigerator 2 Fig 51 pliance may be regaided as equivalent to the combination of a motoi 01 heat-engine dnvmg a lefngeiator 01 heat-pump (fii> 51) A quantity Q of high-tempeiatuie heat goes in al one place, and theieby causes a quantity Q 2 f low-tempeiatuic heat lo go in at anothei place Heat is i ejected at the mtei mediate icinpciaLuie T 15 and the heat so rejected is equal to the sum of Q and Q 2 , foi no woik is done by the appliance 01 spent upon it, as a whole This descuption applies \vhethei the appliance is actually a mechanical combination of a heat-engine with a heat-pump, or is an absoiption machine with no com eision of heat into "uork and woik into heat. In cithei case we have to considei what is the ideally gicatcst lafoo of the low temperature heat Q 2> which is extiacted fiom the cold body, to the high- temperature 01 driving heat Q, when the thice Icmpeiatuies T z , 2\, and T are assigned. '] THEORY OF REFRIGERATION 165 Suppose, first, thai the machine consists of a peifect (icvcrsible) iat-engrne duvrng a perfect (reveisible) heat-pump. Then it is isy to calculate the lalio of the heat extracted Q 2 to the heat ipphed Q Wilting W foi the heaL-equivalent of the work de- 'loped in the heat-engine and employed to drive the heat-pump, e have (by Art. 38) HIT T \ w " ^ i' " ~ T ice the heat-engme is reveisible. Again, since the heat-pump is so reversible, n IT T \ rr/_ ( &U i - ^2) T 2 .Art. 100 Hence = -_) Inch gives the required ratio of heats The impoitance of this result lies in the fact that no other cthod of applying heat to produce cold can give a higher ratio Q 2 Lo Q, the Llnee tcmperaLures T, r l\ and T 2 being assigned. ) pio\e this, imagine the combination of reversible heat-engine id reversrble heat-pump to be reversed rt will then gi\e out an nount of heat equal to Q to Ihe liot body and an amount equal Q 2 to the cold body, and iL will take in an amount equal to + Q horn the rnLcimcdiale boch at r l\ It Mill still develop > \\ork as a \\hole, noi ie<iimc \unk to be spent in dining it. lagme liutlrei thai between the hot bod> and the cold one theic e two appliances vujikmg both using the same intermediate mpcialuie one ol which is this icveiscd combination and the hei is a Kliigc'ialiug machine (such as an absoi phon machine) lose elhcKncy \\e wish to compare wilh lhal ol Ihe combination. u-n il it \\cie possible for that machine to ha\ c a higher enicicncy an Ihe combmalioii, it uould exliacl more heat than Q 2 fiom e cold body for Ihe same expenditure ol high-lcmpcialure heal lie nee, when both work together, namely the combination Diking i excised and the othci machine working direct lire cold idy \\oulcl lose heat while on the whole the hoi body would lose lie In olhei woids we should then have an impossible result, nicly a simple transfer oi heal, by a purely bclf-aetmg agency, >rn Ihe cold body al 7' 2 lo a wanner body at J\, the intcr- scliate tcmperalmc. The agency would be self-acting in the sense being actuated by no form of energy, mechanical or thermal ich a result would be a violation of the Second Law (Art 31) le conclusion is that no means ol employing heat to produce cold, ]66 THERMODYNAMICS [en whether dncctly, as in .in absoiption machine, or milnocLly as in a compiession machine dm en by an engine, CMII be moio efficient (foi the same thiee tempeiatuics) than the combination of a reveisible heat-engine dining a icveisiblc heat-pump Hence the expi ession q^ T 2 (T1\] measuies the ideall}' gieatest latio of heat ex Lidded, to heal supplied. Any real appliance will show a smaller heat ralio in consequence of irreveisible featuies in its action The action of an ammoma-and-water absoiption machine, for example, is very far fiom being leveisihle the heat ratio in iL is much less Ihan unit} 7 -. But, as the above expi ession shows, Avlicn T, is no I much lower than T 1 and T is much highei, Q> may be much greater Ihan Q in the ideal use of heat to produce cold 120. Expression in Terms of the Entropy The above ex- pi ession foi the ideal perfoimance under reveisible conditions may be written Q (\ H_ , rr\ rr\ \ ML2 / rn m \ f 11 fiom which --7ir T 1 2 This expiesses the conseivation of cnliopv lf>i HIP complc-le ie\eisible opeiation The entiopy of the system as u wliolc does not change. Foi the teim on the left is the gam oi entropy by Ihe body at T t to which heat is i ejected the l\\o leims on Ihe nghL are the losses of entiopy by I he hot bod} and cold body icspcc lively The whole action may be legarrled as a tiansfei ol entiopy fiom two somces at T and T 2 to an in bei mediate sink <il 7\. So long as the action is leveisible this tiansfei occms without affecting the aggiegate entiopy, but if it is not completely reversible the ag- giegate entropy will increase m that case the teim on Ihe lei I becomes gicater than the sum of the teims on UK n^lil. Again, the equation shows that, under rcvcisible eondilions, Ihe pioductof the eutiopy lost by the hot somce (Ihiough I lie lemoval of the heat Q) into the diop in tempciatmc Avhich that heul undei- goes, namely fiom T to T lt is equal to the pioduel of Lhe entiopy lost by the cold body into the use of tempcuitme of the abstracted heat Q 2 . Each of these pioducts is in fact a measure of W, the woik which the heat-engine pioduces, and the heat-pump consumes, m the ideal combination of reversible engine with icvci&iblc pump. '1 THEORY OF REFRIGERATION 167 A mechanical analogue is illnslialed in fig 5'2 Heie a quantity ' vvatei M, supplied at a high Jevel H, descends to a lower level ! and serves to raise another quantity M z fiom a still lower level 2 up to II \ Both quantities are discharged at the level H r The relation is reversible, and the eneigy equation may be written M(TI- 7/j) = A/ 2 (7/j - //,) /M H Fin r>2 Miclianinil imAlojriio of I hf usr ol lic.it lo proilncc' cold hi com pa nng I his wilh Lhc equation gi\ en above, i'oi a coiresponcl- i" reversible Iheimalopciation, it will be noticed that the analogue f weight (ol water) is not heat but entiopy, namely the quantity I' heat divided by Lhe tcmpcifituic ol' siipplv. The levcisible llieimal opeiation may be lepicscnted on the nli'opy-tempciatuic diagiam as m (ig 53 There the area abon L'prcsciiLs the high-temperature heat which is supplied at tcm- cvatuic T, and the area fibcd icpic&cnts the work which would 168 THERMODYNAMICS [en. be done in a peifect heab-engme by letting clown that quantily oi' heat from T to the lower level 2\ Between the given levels of temperatuie T a and T 2 diaw a icctangle dcfg whose aiea is equal to the aiea abed, and pioduce cf to meet the base line lor zeio tcm- peiatuie m m. Then the area fgnm icpiescnts the refugeiatmg effect, namely the heat exti acted fiom the cold body at 2' 2 ' The m no Fig 53 amount of heat dischaigccl at the in tei media k 1 level T^ is equal lo the aiea ecom, which is equal to the sum of the aieas abon and Jgnm 121 The Refrigerating Machine as a means of Warming. In any such appliance, whether levcrsible or not, the quantity of heat delivered at the intermediate tempeiatuie 'J\ is greater than the quantity supplied at T by the amount of the heal uiisccl fiom T 2 , and may, as ue ha^ c seen, be much giealei This fact is the basis of an mteicsting suggestion made b} Kelvin in 1S52, that in Ihe warming oi looms it would be Iheimally moic economical to apply the heat got fiom binning coal in this indirect way than to discharge it into the room to be waimcd. The thcimody mimic \ alue of high-tempeiatine heat is wasted if we allow it dncetly to entei a comparatively eold substance That value might be better utilized by employing the heat to pump up moie heat, taken in horn say the outside atmospheie, to the level lo which the room is 1 THEORY OF REFRIGERATION 169 be warmed By using, foi example, an efficient steam-engine chive an efficient heat-pump, a small quantity of heat supplied a high tempeialure will suffice to laise a much gieatei quantity heat thiough the small range that is requned, and consequently produce a much gi eater warming ellect. Similarly, if a supply ' power fiom any souice is available as a means oi warming to moderate tcmpeiatme, it will be tinned to bcttei account if we t it to duve a heat-pump than if we simply conveit it into heat, he suggestion that some of the coal which is used loi heating >oms might be saved by applying heat in this indnect mannei is at piesent no more than a theoietical mtciest. 122. The Attainment of Very Low Temperature, Cascade lethod Anothei pait oi the science oi lefngeiation deals with icthods of pioducing cold so exticme as to hqueiy an and other ^-called pcimanc-nt gases This is now the basis oi an impoitant iclushy, which employs the liquefaction of an as a step to ^\ a ids ic sepaiation of its constituents, with the object oi making com- icicial use oi the oxygen 01 the mtiogen or both To liquefy ny gas the lempeialme must be icduced below the cntical point Vit. 77), and foi mtiogen this moans a cooling below 110 C 'empeialuics much lowci than thi^ have been i cached by the K' I hods \\luch \\ill now be descnbcd Hjdiogcn, whose cntical :mpeiatine is 211 C , has not only been hquehccl but M>hdilicd s melting [joint unelei atmospheiic piessnic is about 25S 3 l. i 15 absolute E\ en helium, tlie most leiiactoij ol all known ases, has been liquefied nuclei conditions that lo\\ucd the' cmpcialinc to within I luce or i'oiu dcgiccs oi the absolute /eio One uay oi leaching a VIM low lempeialme, called the "cascade" iicthod, is to have a seues oi compicssion uJngeiating machines o connccUd that Ihc \\oikmg substance in one, when cooled by ts own cvapoiatiou, acts as the cnculatmg fluid lo cool Ihc ondcnsc'i oi the next machine ol the SULC.S, and so on Difleient \ r oikmg (luids aie selected loi the successive machines, so that ach in tin n ic aches a lowci lempciatnie than its picdccessoi The gcncial idea oi the method is illustiatcd in iig. 5J In that hagiam the lust woiking substance is caibomc acid, which is cpicsentcd in the sketch as supplied from a icservoii on the left, nlo which it has been compressed It expands thiough atlnottle- iilvc into the vessel A, fiom which it escapes at atmospheiic nessuie (this pait of the apparatus might be completed by a 170 THERMODYNAMICS CTT compression pump restonno the substance Lo the icscrvoir). The effect is that the vessel A is kept at a temperatuic of abouL - 80 (' Within it is anothei vessel which ser\ r es as the condenser of d machine using ethylene as working substance Ethylcnc has a cutical tejnpcrature of 10 C., and needs only a modciatc piessnie to liquefy it at 80 C. It is pumped into the inner pait of the condensei A, is thcie liquefied, and passes on through an expansion-valve to the outer part of the vessel B in which if eva- porates, pioclucmg a tempeiatme of say 130 C al the low pressure which is maintained by the pump. Thus cools I he vessel B below the critical point of ox}'gen (namely 118 C ), accoid- Ethylene Oxygen -80C -130 C -200 C Fig 54 Cascade Method of leaching veiy Low Tempeiatmoq mgly oxygen may be used as the woikmg substance of I he next machine It is condensed m the mnei part of the condenser Ji, and aftei passing thiough an expansion-valve it may pioducc a tenipciaturc of - 200 C or less in the vessel C by cvapoialmg theie nndci alowpiessmc Each machine of the sencs is a vaponr- compression machine, vioikmg on the pimciple already dcsciibed, and made up of an evapoiatoi, a compicssmg pump, a condensei, and an expansion- valve The essential featuie m the combmahon is that the woikmg substance in any one machine must be e\a- porated at a tempeiatme that is lower than the ciitical poml of the woikmg substance of the next machine m the scucs. v] THEORY OF REFRIGERATION 171 123. Regenerative Method Bui it is in a dificicnt way that ow tcmpciatmes arc now attained for the commercial liquefaction f an. The usual pioccss is a regenerative one, first successfully levcloped by Lmde, in which the Joule-Thomson effect of nrever- ible expansion in passing a constncted onfice serves as the step- lown in tempera tin c, and a cumulative cooling is produced by ansing the gas which has suffered this step-down, to take up leat in a thermal inteichangei from anothei poition of gas that s on its way to the ounce. Consider first what would happen if theie weie no such theimal nterchange. Imagine a gas such as an to have been compiessed a high prcssuic P i, and to have had the heat developed by com- nession icmoved by circulating watei 01 othenvise, so that its empeiatuie is that of the suuoundmgs. Call this initial tem- )eiatuic TI Let the compiessed gas at that tempcratiue enter an ippaiatus in whicli it expands irreveisibly (thiough an expansion- 'alve 01 plug or constricted ounce of any kind) to a much lowei HCSSIIIC Ph, at which prcssuic it lca\ es the appaiatus If the gas vcie an ideal peilccL gas this nie\ cisiblc expansion would cause no all in tempciatiuc In a ical g tl s Iheie is in gencial a fall, fiom C l to some lowci tempciatuic T. The fall 2\ - T mcasmes the Foule-Thomson cooling effect of the gi\ en chop in picssuie. In Foiile and Thomson's cxpciiments on an it was about a quailei )l a clegue foi each atmosphcic of chop in picssuu ! Tlic cooling c fleet of the chop in picssmc may he measuied by he quantity oi heat which uould \\&\ c lo be supplied to the gas, )ei Ib , afici expansion, to icstoit it to the tcmpciatme at which 1 cntcicd the cippaialus Call thai quantity Q then f\ T tT 1"\ Q = A ,i (- 1 1 - * h vlieic A',, is the mean specific heal ol I he gas between these tern- K'latuics, aL the lowei picssuie P,, We may define Q as the quantity of heal which each pound of the as would liavc to lake up within the appaiatus if iLs Icmpciatuie m lca\in Ihc nppaiaLus wcic made equal to its leinpeialuic on uliy It measuics Hie available cooling effect due to each pound )f gas llial passes thiough the apparatus. * Ac Lending to then icsults loi an, Uio fall of tompcratino oxpiossocl m degiecs cntigraclo is /">7'i\ 2 0275(P,- P)(^). vhcuo P i and Pjt arc tho piossmes in atmoflphoi cs 172 THERMODYNAMICS [en So long as there is no communication of heat to the gas, by theimal mtei change 01 othei wte, while it is passing thiough the appaiatus, the gas simply passes off at a lower tempeiatine T'. The gas that passes off has the same total heat I as the gas that enteis (Art 72), though its tempeiatme has chopped If we were to restoie it to the ongmal tempeiature T before letting iL pass oft, it would take away moie total heat than it bungs in, the cliffei- ence being equal to Q. Its total heat / at exit would then be gi eater than its total heat on. admission by the quantity Q, though its tempeiatme would be the same. The existence of a Joule-Thomson cooling effect in any gas depends on the fact that I he 1otal heat 1 is a function of the piessuie for a given tempeiatme the total heat is gieatei when the piessure is low. Suppose now that theie is a counter-cm lent mteiehanger by means of which the stieam of gas which has passed the oufice takes up heat from the stieam that is on its way to the ounce, with the result that the outgoing stieam, before it escapes, has its tem- peiatuie lestored to T or veiy near it. Tins is easily accomplished by having, within the appaiatus, a long appioach pipe or woim through which the compiessed gas passes befoie it i caches the ounce, and lound the outside of which the expanded gas passes away, i>o that theie is intimate theimal connection bet \\cen the two stieams. Foi simplicity we may assume the inteichangei to act so peifectl} that when the outgoing gas leaches the exit il has acqmied the same tempeiatme 2\ as the entenng gas Each Ib of it will theiefoie have taken up a quantity of heat equal to Q. as defined above. 124 First Stage Undei these conditions the appaiatus will steadily lose heat at the late of Q units foi eveiy pound of gas that passes through If we suppose the appaiatus as a whole to be theimally insulated against leakage of heat into it from outside, theie will consequently be a continuous i eduction of the stock of heat that is held by the pipes and the gas in them The lesult is a piogiessive cooling which constitutes the hist stage ol the action It may help to make the action cleai ]f we chaw up an account of the eneigy icceived and dischaiged bv the apparatus Gas enteis at A (fig. 55) undei the piessuie P / and at the tempeiatuie T! Gas leaves the appaiatus at B under the piessuie P n and at the same tempeiatme T" l5 having taken up, through the aclion of the mteichangei, a quantity of heat equal to Q. The pipes and iv] THEORY OF REFRIGERATION 173 expansion orifice arc noL shown in the sketch they aie within the enclosing case, which is assumed to be a peifcct non-conductoi of heat. Dining the fiist stage of the action the stop-cock C is closed, and all Lhc gas that has gone in at A goes out at B, it is onh T \)y the cntiy of gas al A and by its escape at B that eneigy enters 01 leaves the appaiatus Each Ib of en Leung gas contains a quantity of internal eneigy E i, and the woik that is done upon it as it goes in is P^V A Piach Ib of outgoing gas contains a quantity of internal eneigv EU, and does woik, against external picssme, equal toP/J 7 /; Hence, for each Ib that HOAVS thiough, the net amount of heat which the appaiatus loses is ^ a + P n V a -(E l + P l V. l ), 01 / Z ,-I, But the amount so lost is Q, namely the heat that is lequued to tcstoic the gas to the lempeiatuie at which it makes its exit. Hence Q = In 1 1. Fit/ 55 The contents ol the nppaiatus become coldci and colrlci in "* jonscquenec ol this conhmud fibsliaclion ot heat But it is nn- soilant to notice that Iheu fall in tempoialiuo does not affect he value of Q We assmm thai Hie action ol the lluimal inlci- hangci continues to be prilect, in that ease llic exit tcmpeiatuie ,vill still be e([iial to the initial tcmpcialuu 7\ lio^c\ci cold the nteuoi becomes in the ncighbouihood ol the expansions alve. riieie will be no change in the \aluc ot either T ]S 01 /,, and conse- jucnllv no channc in Q The value of Q, as the abo-sc expicssion .hows, depends cntnelv on the conditions at A and at B, with KM feet interchange this means that it depends only on P t , PJI, and f\ It is independent of any tcmpciature conditions within the ippaiatus It is thcicfoic not affected by the piogiessivc cooling, ind ictains the same value as the action pioceeds*. * It will bo nhown in C'haptoi \ II that tho quantity Q, which moasuics the vailablo cooling efloot within the apparatus -when the pressuics P ( and P ]t and dmission tcmpfiatuio T\ aro asaigned, can bo calculated jf wo know the coefficient 174 THERMODYNAMICS [en. This staoe of progiessive cooling continues until the temperaluic of the gas at the place wheie il is coldest, namely on the low- pressuie side of the expansion- valv e, falls not only below the ci itical point, but to a value T 2 which is low enough to let the gas begin to liquefy undei the pressiue P B . In othei woids T>is the boiling point coriespondmg to PB This is the lowest tempeiatiuc that is leached Fig 56 Fig 56o Ideal pi ocess ot Regeneiativc Cooling A continuous giadient of tempeiature has now become csLab- hshed along the flow-pipe within the appaiatus fiom Iho point ol enliance, wheie it is 2V to the high-piessuie side of I he cxp.msion- \alve, wheie it exceeds T 2 by the amount of the Joule-Thomson of expansion of the gas under constant pieasuie foi v.uious piossuios, and also flic volume (pei Ib ) for various pictures, at the tompeiatuio T t Wilting V ii i Mm volume at any pressure and (^) foi the coefficient of expansion, namely Mm iate of change of volume pei unit of change of constant, we shall see there (Ait 1S2) that mo when tho picssuio IN the temperatures being taken as T l thioughout Q may also be found e^peiimentaUy, bj observing tho chop ol lompomlmo li - 1 which takes place when the gas expands fiom P L to P, t thiough a Joulo- ihonison orifice without any interchange of heat r] THEORY OF REFRIGERATION 175 L-op. Theie is also a conbinuoiis gradient along the retain pipe onr T 2 , on the low-pressuie side ol' the valve, to 2\ at the exit he flow and leturn sti earns aie in close thermal contact, and at ich point theic is an excess of temperatme in the flow which lows heat to pass by conduction into the leturn, except at the itiancc Avheie, under the ideal condition which we have postulated ' peifcct inteichanqe, the tenipeiatiue of both floAv and retinn Z'i Tint, state of things is diagianrrnatjcally represented in fig. 56. heic the flow and letuin aie icpiescnted as taking place in stiaight pes, one inside the othei to piovide loi interchange of heat, nlering along the innei pipe A the compiessed gas expands nough a constricted onfice E (eqim alent to an expansion-vah e) to a vessel fiom which it letinns b}' the outei pipe B The 5 sscl is piovidcd with a stop-cock C by which that pait of the fluid Inch is liquefied can be diawn ofl when the second stage of the )eiation has been i cached In the tempeiatuie diagram (fig 50 a) r N represents the length of the intcichangci, DM is the initial ncl final) tenipeiatiue '1\, GN is T>, and FG is the Joule-Thomson op. DF is the giadientfoi the [lew-pipe, and GD foi the letuin. 125 Second Stage When this giadicnt has become established egds begins to hqucf) , the appaiaLus does not become any coldei, id the action enters on the second stage, which is one of thcimal [iiilibimm A ceiLam small iuiction ot I he gas is continuously juc'hcd and may be dunned oil as a liquid Uiiough the stop-cock The' laigc'i iiaction, Avhich is nol hquclied, continues to escape longh Ihe inlciehangci and to lca\ c (he appaiatus al thr same mpcialinc as befoic, namely the tcnipciatine r l\ equal to that Ihe enU'iing gas Call this unliqiiefied fi action q, the'n 1 q picscnts the fi action that is eliawn oil as a liquid at the tcmpcra- ie j 7\ Since lire apparatus is now neithci gaining noi losing til on Ihe Avliolc, ils heat-account must balance, horn Avlnch IA = ql,> + (1- 0)/ t , icrc /j( is the total heat pei Ib of the gas entering at A, IK is the tal heat per Ib of Lire gas leaving at B, and /,- is the total heat r Ib of the hqurd leaving at C In this steady Avoiking the gregate total heat of the fluid passing out is equal to that of the id passing in. The fluid, as a whole, takes up no heat in passing rough the appaiaLus 176 THERMODYNAMICS [en. Suppose now that the liquid leaving at C were evaporated at its boiling point 7\, and then heated at the same piessure fiom 2\ to Tj. The heat required to peifoim that opeiation would be (1 - q) [L + K, (2\ - TJ\. But that hypotlictical opciation would icsult in this, that the Avhole oi' the fluid then leaving the apparatus would be icstoicd to the tempeiature of entry, namely 2\, sinee the pait which escapes at B it> already at that tempeiatme Hence the heat ic- quned for it is equal to the quantny Q as defined in Ait 123 We theiefoie have Q. from which l-q= ---, This equation allows the fraction that is liquefied to be calculated when Q is known' The fiaction so found is the ideal output of liquid, for we have assumed that theic is no leakage of heat fiom without, and that the action of the inteichangei is peifeet in the sense that the outgoing gas is laised by it to the tempciatuic of entiy. Under leal conditions theie will be some theimal leakage, and the gas will escape at a tempeiatme some\\hat lower than T 1 the effect is to diminish the fiaction actually liquefied The fiaction 1 q is mcieasecl by using a laiger piessurc-diop. It is aKo mcieased by i educing tlie initial tempeiatme T I} thus the output of a given appaiatus can be laised by using a sepaiatc lehigeiatmg device to pie-cool the gas Pie-cooling is indispensable if the method is to be applied to a gas m which, like h> drogen, the Joule-Thomson effect is a heating effect at oidmaiy tempciatiucb, but becomcb a cooling effect A\hen the initial tempeiatme is sufli- cicntty low. 126 Linde's Apparatus. The punciple of legeneratn c cooling descubed m the piccedmg aiticle was fiist succcsslully applied by Lmde m 1S95 foi the pioduction of extiemely IOA\ temperatures, and for the liquefaction of an, by means of an appaiatus shoA\n diagiammatically m fig 57 It consists of an inteichaneer CDE foimed of two spiral coils of pipes, one inside the othei, enclosed m a theimally insulating case. A compiessmg pump P dchveis an under high piessuie thiough the valve H into a coolei J Avheie * The specific heat of the vapom is horo treated as constant from T z to T I} which is very neaily true at low pressures THEORY OF REFRIGERATION 177 he heal developed by compiCbsion is lemoved by water circulating n the oidinaiy way fiom an inlet at K to an outlet at L The uglily compressed air then passes on through the pipe HC to the imcr woim and aflcr travel sing the woim it expands through the hioltle-valvo ft into the vessel T. Iheieby snffciing a diop in cmpciatuic. Then it lelurns thiougli the outer woim F and, being n close contact with the innei woim, gives up its cold to the gas hat is slill on ils \\ r <iy to expand Finally it icachcs the com- ncssion cyhndei P through the suchon-\ alve G, and is compressed o go again Ihiough Ihc cycle Dining the first stage it simply goes ound and round in this way, but when the second stage is i cached iul condensation begins, the pail that is liquefied is diaunoff at V nd I he loss is made good by pumping m moic an tluough the stop- alvc al A by means of an auxihaiy low-picssurc pump, not shown Fig f~>7 Limb's llo^enoialuo Ap|)<uutii^ L the sketch, which dchveis an lioiu the almospheic to the low- lessiue side ol the ciiculatiiig system Liude sho\\ed I ha I bv keeping this lowei picssuic fanly high, il [)iactieable to icducc the amount of uoik thai has to be spent hquel\ing a gi\en quantity ol air He pointed out that while ic cooling clled of expansion depends upon the diffeicnce of essuus PI and Pn on the two sides of the expansion-valve, the oik done in compressing the an in the ciiculatmg system depends i the ratio of /' / to P/ ; . Il is loughly piopoi lional to the logarithm ' I hat witio, (01 il approximates to the woi k spent in the isothermal mipicssion of a pcifecl gas, which (by Art 28) is 7?Tlog t ?, here ; is the lalio of the volumes or of Lhc piessnics. If, for :ample, P., ib 200 atmospheics and Pn is one atmosphere, the lolmg ell'cct is proportional to ]99 and the work of the compiessmg imp is 10 uglily proportional to log 200. If on the othci hand the 13 T 12 178 THERMODYNAMICS |c-ir back piessiue Pn is 50 atmohphcics Ihc cooling elTicl is piupor- tional to 150 and the woik of the main compicsMng-pump (o log I The cooling effect is icduced by only abonL onc-lbmlh, while Ihc work is leduced by nearly three-fourths ALLci allowing I'm Uu i extra amount of work that has, in the second case, lo be s|>enl on the auxiliai}^ pump in supplying an at 50 nlniospheies lo replace the fraction which is liquefied, there is still a marked adxanlage, in point of theimodynamic efficiency, m using a closed cycle \\ilh a modeiately high back piessure. The Lmde process is employed on a commeicial scale lo liquefy air as a first step in the separation of its constituents A Linde plant at Odda, m Norway, liquefies about one hundied tons ol air daily for the pmpose of supplying nitiogen I'oi use- in I he inanu- factuie of cyanamide, an arhlkial nitrogenous t'erlili/ei \\Iueli is formed by passing gaseous nitiogen over hot calcium cui bide The method by which the constituent gases aie separated will be presently desciibed. 127 Liquefaction of Air by Expansion in which Work is done. Claude's Apparatus. The chop in lunpualme which a gas undeigoes in parsing fiom a legion of high piessmv lo ,1 i< uion of low piessuie would be gieatcr if Ihc piociss \u-i< eniiduclcd ie\eisibly, as b> expansion m a cyhndci in whuli llu- "as ( | l)OS mechanical woik We should still luu c Ihe small Joule-Thomson cooling effect, but in addition thcic would be Ihc (n t nei.illy much laigei) cooling effect that is clue to the eneigv \\lueh (lu i>as loses in doing woik Eail> attempts made by Siemens, Sol\ ay, ami olhc is toieacrMery low tempeiahnct, by applying a lheiiu.il iiileich.innci to an expansion cylinder, failed mainly because the e \hude, S.MMI leached a tempciatuic at which the lubiicaul ho/e Tins dilliriilh ^as successfully oveicomc in 1902 by Clamh , u ho (oiind Ih.il I he difficulties attendant on expansion m a woilunn cvluulf i ,|, mn to a tempeiaiuie below the cnt.cal point of an could be ove u-omc by using certain hydiocaibons as lubucants A I M (Imc.iilxui s,,( h ,s petioleum-ethei does not solidity but rcm.uns N ,scous nt u-iu- peratuie as low as - 160 C Usm a a lubueant of llus lund (l, 1K Ie succeeded, as an expcumcntal low dc (<ncc, ,., ], ( ,,,ef\'m.. an- , an expansion cylmder furnished wilh a icgemn.hve rmmh,- cuiient thermal mteichangei the cxpaus.on cyluule, snnnly taking the place of the expansion-vnl V eMn ana 1 , 1 )arnlussuchas Ihal of Art. 120. He also found that Ihc liquid, once it h, ms to fo, n IV] THEORY OF REFRIGERATION 179 serves itself ns a lubricant, and no other need then be supplied Under these conditions, however, there is little if any advantage in using an expansion cyhndei, for the volume of the fluid at the lowest exticme of tempeiatuie is so small as Lo make the woik of expansion insignificant. Theie is not much additional cooling, at the same lime it is far less practicable to secure theimal insulation with an expansion cyhndei than with a Joule-Thomson ounce. Claude subsequently obtained a more economical icsult by giving the appaiatus the modified foim shown in fig. 58 In that airange- nicnt pait of the compressed air expands in a woikmg cylinder to a tempeiatuie which may be just below the critical temperatuie, and the an which is cooled (but not liquefied) by that expansion is used as a cooling agent on the remaindci of the an, with the csull lh.it soim- o( the lalUi is liquefied nuclei the higher pussiuc il \\luch it is supplied The stipph comes in, al a picssiue of 40 ilmosphcics 01 so, lluough the eenlial pipe of Ihe coimlei-cmient nlc'icluingci M Pait of it passes mlo the expansion C3 r lmdcr D vlit'ie it expands doing work, and is then chschaigcd thiough the iondcnsmg \ essel L, wlieieil sei \ es as the cooling agenl lo maintain i lempeialiuc .somewhal Ie)wei than H0 C., the cntical tcm- K inline ol an The lemaindei of the eompicssed an enters the nbcs ol L and is condensed thcic, undci piessme, chopping as a iqmd mlo Ihe chambei bclo\\ fiom which it can be chawn off In ji I'nilhcr development of Uns invention Claude made the 'xpansion compound, and caused the expanded gas to act as a ooling agent after each stage, becoming itself \varmed up in the >roccss The expanded gas is thereby picpaied to suffer further 122 ISO THERMODYNAMICS [en expansion without an excessive fall of tcmpeiatuic. Dmmg Us expansion the gas in the C3 r linder is not so neai the liquid state as to make expansion in a woikmg cylinder of hi tie use The arrangement with compound expansion is illustiated in fig 59. An undei picssuie enters, as before, tluough the central pipe of M. Part of it goes to the fnst expansion cylindci A, docs ivoik there, and piocceds at i educed prcssme, and ata tempeiatuie below thccnlical point, thiough the outer vessel of the condenser L lt in themnci tube ol which some of the compicssed an is being condensed This waims up the expanded an to some extent, and it then passes on to complete its expansion m B, which again brings its tcmpcialine down sufficiently to allow it to act as condensing agent for the Fig 59 Claude's Intel method Aulh compound expansion remaining poition of Ihc an under piessine, in I he second con- dense! L 2 This division of Ihe expansion into I wo (01 il nuiy be moie than two) stages is equivalent to making Ihe piocrss as a whole more nearly isolhcimal, so IhaL Ihe an need not al any st.igc deviate veiy widely fiom a tcmpeiahuc which is just sulhcicnllv below the critical point to allow liquefaction to go on undei Ihc pressure at which the an is .supplied 1 ' 128. Separation of the Constituents of Air. The lique- faction of an enables the constituent gases to be sepaiatcd because * G Claude, Gomples Rcnrlus, II June !')()(>. and 22 Of, 1000 Hen H!HO Jus book, on Liquid An, Tis f'oUioll, 191 { An iuhclo by PJLOOHSOI E Matluas in RcnLc. g&t&ale dcs Si icnccs, 15 SopL 1907, contains un mtcioatmg aocounb of ilu> wliolo subject of ilie indusbnal liquefaction ot au v] THEORY OF REFRIGERATION n ic evapoiation they have diffeient boiling points The boiling DOint of mtiogen, under atmospheric pressure, is about 195 C n 18 lower than that ot oxygen, which is - 182 C. When a quan- ,ity of liquefied an eAapoiates fieely both gases pass off, but not n the original piopoihon in which they aie mixed in the liquid The mtiogcn evapoiates more leachly, and the liquid that is left jecom.es iichei in oxygen as the evapoiation proceeds This lifference in -\olatihty between oxygen and nitiogen makes it possible to cany out a piocess of rectification analogous to the jiocess vi Inch is used by cbstillcis foi extiacting spuit ii^m. the 'wash" 01 lei merited wort, which is a weak mixtuie ot a nhol ind \vatci, by means of a device known as the Coffey Still In the still patented by Aeneas Coffey in 1830 theie is a rectifying jolumu consisting of a tall chambei containing many zig-zag shelves 01 bafllc pLites The wash enteis at the top of the column ind tuckles slowl} down, meeting a cuiient of stearn which is idnuttcd at the bottom and use-, up thiough the shelves. The lown-commg wash and the up-going steam aie theieln brought into close contact and an exchange of fluid takes place. At each -.tagc some of the alcohol is e\apoiated fiom the \\ash and some of the steam is condensed, the heat supplied b\ the condensation of UK steam seivmg to e\apoiate the alcohol The condensed steam becomes pa.it of Uie do\\n-coming sticam of liquid the e\ apoiated ilcohol becomes pait of the np-gomg sticam of \apom Finally it llic top a Acipour compaialix eh nch in .ilcohol passes off at the l)oltom a liquid accumulates \\liich is watci \\ith little 01 no ilcohol in it A tcmpualuic giadicnt is established in the column it the bottom Ihc Umpeiafuic is llmt ol steam, and at the top Ihc'ir is a lowei knipcialiuc appioxunating to the boiling point it alcohol The wash enti-is at this compaiatively low tempeiatuie, ,uul lakes up heat horn Iht steam as it tnckles down Lnidc applied the same geneial idea in a device foi bepaiating Lhc less volatile o\\gen horn the moie volatile nitiogen of liquid in In this de\ice, the pnmaiy puipose of A\ Inch \\a^ to obtain Dxygen, theie is a iccti lying column down vihich liquid an tnckles, -,Laihng at the Lop at a tempeiatuie a little undei - 19J. C 01 70 absolute, \\luch is the boiling point of liquid an nuclei at- mosphei ic picssmc As the liquid tiicklcs down it meets an up-going aream of gas which consists (at the bottom) of neaily puie oxygen, initially at a tcmperatuie of about 91 absolute, that being the boiling point of oxygen undei atmospheiic pressure As the gas 182 THERMODYNAMICS A A uses and comes mto close contact wilh the clown-coming liquid, theie is a give and take of substance at each shii>(' some of the using oxygen is condensed and some ol Ihc in the down-coming liquid is evaporated, the liquid also be- comes lather wannei BY the time it i caches the bottom iL consists of neaily pure oxygen the nitrogen has almost com p 1 et cly passed off as gas, and the gas wlir i passes off at the top uon- SIL jveiy largely of mtiogen Moie piecisely it consists of mtiogen mixed with about 7 pei cent of oxygen in other woicls, out of the whole oiigmal oxygen content of the au (say 21 pei cent ) two- thuds aie biought do\\n as liquid oxygen to the bottom of the column, while one-thud passes off unsepaiated along with all the mtiogen The oMgin that gatheis at the bottom is with- drawn foi use, and is c\ apoiatcd in sen ing to hquefv fiesh com- pressed an, which is pumped into the apparatus to undergo the pi o- cess of separation The cold gases that aie leaving the ap- paratus, namely the oxygen which is the useful pioduct, and the mtiogen which passes off as wastegasatthe top of thecoluirm, aie made to ti averse counLer- cunent mtei changers ou then way out, so as to give up then cold B F C) '////////////777T?//'/'//', l fl)I to the incoming compressed mi ]PIR ()0 LimItfllll|lpllMilllH , that is on its way to be liquefied o\inuLin{- o\y^n I>y icdiduiiiou In the chagiam, fig GO, these conn I ci -cm ion I mloic lumens ,11 1 omitted for the sake of clearness, bul I he csst-nlial Icaluus ol llu 1 condensing and lectifymg apparatus are shown The (igiue is v] THEORY OF REFRIGERATION 183 Dased on one given m Lmde's patent of June 1902, which descnbes he m\cntion by which a process* of lecLiiicalion has been success- 'ally applied in the ex ti action of oxygen fiom an. The ic A is the rectifying column, consisting m tins instance of i vcitical chamber slacked with glass balls, through the mteistices )f winch the liquid tuckles clown The lowei pait B contains an iccnmulation of fluid which, when the appaiatus has been at work ong enough to establish a unifoim legwie, consists of neailv pine iqnid oxygen Compressed an, which has been cooled by passing Jnough a co untci-c uncut mtcich.ingcr, entcis at C, becomes iqucficd in the veitical condenser pipes D, which aic closed at he top, and diops dovn into the vessel E It gnes up its latent icat to the oxygen in B, thcieby evapoiatmg a pait of that, and ,o supplying a sticam ol gaseous oxygen which begins to pass up he icclif>mg column. On its wa\ up, this stieam of gas effects in exchange of maUual with the liquid an which is tiicklmg lo\\n gasious oMgen is condensed and leluins with Ihe sticam o the \ esse I B, while mtiogen is (.\apoialcd and passes oil at the op ol I he column, al A r , mixed \\ilh some o\\gen The escaping jas goes lhioiiL>li an inlciclumgci, taking up IK at liom the m- ommg compiessed an The a cci mml.il 101 ioL neaiK pine liquid oxNgcn m//o\ cillows into lu lo\\ci vessel F, wh< ic a suppkniuilai \ supply of compiesscd an nUimg al (1 is employed to <_\apoiale it In me, ins ol a siimlai u languni ill ol condense i lubes open al the bollom and closed il UK lo]), this .in bccominii ilsell condensed in I he pioci'ss, anil allmg as a liquid into Ihc \esstl // The limited an fioni I* ind tiom // is slill undiM picssiiit il passes up tliiouyli t \pansion- , ,il\ es K lo I lie lop ol Ihc icclilving column, win le il is dischaiged >\(i UR ylass balls al ii pussiiu nol makiialh abo\L thai c>( the ilmosphuc This secmcs llu incessai} dilftuncc in k-nipiia- inc bclucen Ihe bottom and top of the column The com- >tisscd an l)la^s Ihe pail ol hc.itci and i\ apoialoi ol Ihe liquid )x\gen <if the bollom. at the compaialivcly high lempeiatme of iboul 01 absolute, be loie ituiicleri^ocs uctification Inolhei woids, L nol only coiicsponds to Ihe "wash" of Ihe Cofley still, but it ilso sei\es as the equivalent of Ihc hcatci by which the liquid at [he bollom of the still gi\Cb off an upwaid cuiuat of steam [Jascous oxygen, Ihe dcsiied pioducL in this case, passes off al 0, ind like the waste gas, consisting mainly of niliogen, which escapes ifA T , it goes tluough a counter-cuuent mterchangei, taking up heal 184 THERMODYNAMICS [en fiom the compiessed an which enteis partly at Cand paitly at G It is the waste gas in this pioeess that foum Lhc analogue of the lectified spuit Avluch is the useful pioduct of the Coffcy still. At first, when the machine begins woikmg, Lhe air is highly com- piessed, but aftci the opciation has gone on foi some Lime, and a steady state is appioachcd. a much lower piessme is sufficient IL must be high enough to make the an liquefy at the IcmpciaLuie of the liquid oxygen bath, say 91 absolute, ami in pincticc il is kcpL 100 10 20 30 40 50 60 70 80 PERCENTAGE OF OXYGEN IN LlOLI/D F,g (>1 higher than this to emiuc that Ihe diop in lcni|K ruLmc al Liu expansion -vahe may be sufficient to make good any losses due to leakage of heat fiom outside, and to impelled mlci change in the counter-curient appaiatus Foi sonic time, after I he iippaialns is first staited the icctifying aclion is impel (Vet, bul as I lie pioocss goes on the liquid contents ot the vessel Ji become i ichei and nchei m oxygen, the icctification becomes moie complete, and Ihe piessure maj be icduced Unclci piaelical condilions it is easy lo secuie that the gaseous pioduct shall lie pine lo iho extent of containing 98 per cent, of oxygen ] THEORY OF REFRIGERATION 185 129 Baly's Curves The action of the lectifymg column will * ma.de moic intelligible if we icfer to Lhe icsults oi cxpeiiments ibhshed in 1900 by Baly*, which deal with the nature of the apoiafion m mixtuies of liquid oxygen and mtiogen. Given a ixtiue of these liquids in any assigned piopoition, equihbiiimi tween liquid and vapoui is possible only when the vapoui ntains a definite piopoition of Lhc two constituents, but this oportion is not the same as that in the liquid mixtme Say for ample that tJic liquid mixtuic is half oxygen and half mtiogen, en according to Bah 's cxpeiiments the vapoui pioccedmg from ch a mix lure will consist of about 22 per cent of oxygen and i pci cen I. of mtiogen With these pioportions theie will be [iiilibiium If howcvei a vapoui iichci than this m oxygen be ought into contact Avith the half-and-half liquid, pait of the seous oxvgen will condense and pait of tlie liquid mtiogen ill be cvapoiatccl, until the piopoition giving equilibiium is aclied The cm \ e, fig. (il , shows, toi each piopoition in the mixed 1 1 ud, \\hat is the coiicspondmg piopoition m I lie \ apom nccessaiy i 1 equilibimni in olhei woids wlwt is the pioj)oibou which the nslitucnls luu c in the v.ipoui, when that is beiny J'oimed b"s apoi ilion of the mixed liquid, in the h'isl slayes ol such tin c\ti- >iat ion, befoie UK piopoi lion m the liquid changes In I his cin\ e e base-line specifics I he piopoition ol ox \ gen in the liquid nn\ tin c, )iu lo 100 pel eenl , <ind the oidmales <>i\ c the piopoihon of \gcn in I he coiicspondmg \apom, A\hcn I he \fipom is (oinud ide i .1 piesMiic equal lo lluil ol Ihe almosphcic Mueh I he same, ncial ielalu/11 \\ill hold al olhci piessuies 11 will he seen fiom e eiii'\e' lh.il when Ihe e \ apoialing iKjiud mi\lme is liquid an \vgcn 2^ pel cenl , mliogen 79 pel eenl ), the piopoilion ol yyen piest nl in Ihe \ apom I hat is ( oming off is aboul ? pei cent. a hi lie less. This is what oeeius al Ihe lop of the lecliiVmg column in the )j).uatus ol lig (!() The Juniid lhat is e\ apoitiling Iheie is lieshly imed liquid an, and he net Ihe waste gases cany ofl about 7 pei nl of oxygen Coming down Ihe column Ihe liquid hnels itsell eonlael wilh gas containing moic oxygen than concsponds to uihbiumi Acfioidmgly oxygen is condensed and mtiogen is apoiatecl al each stage in Ihe descenl, in Ihe elloil at each level a condition ol eqmhbiium between the liquid and the wilh wJuch it is there in contact. Ba\y,Plul May ,\o\ xux, p 017, 136 THERMODYNAMICS [cir Fig 62 is another foim of Baly's cuive, the form., namely, 111 which the icsults of the experiments wcie ongmally shown. Thcic the oidinates icpicsent the absolute tempcialmc (m centigrade 30 4O r .O nO 70 PERCENTAGE OF OXYGEN Fig 02 degiees) at \\hich, under almospheiic prcssinc, llu- i in. \c-cl li(|iii(l boils, and two ciuvcs aic dia\\u which show by IIKMM.S ol I he scale on the base-line the percentage consLiLulion ol (1) Ihc li(|iml, (2) the vapom, when the condition ol. cquilibnuni bcLvvtcn lupud THEORY OF REFRIGERATION i 8 7 L vapoiu is attained*. A honzontal line drawn across, the curves any assigned level of tempeiatiue shows the composition of )our and liquid icspeclively foi that tempeiatme, when the two in cqnilibiium Taking an inteimediate point between the top I botlom of the lectil'ymg column, and di awing the hue for coiicsponding tcmpeiatuie, we should find the lespective vposi lions of liquid and ^ipom theie to appioximate to the ucs found fiom Ihe two cunes, this appioximation being closer moie slowly tlie liquid trickles clown, and the moie intimate contact between liquid and gas f a simiLii condition of equilibrium holds at each stage in the cess of liqnclyjng a nuxtuie of the gases, these cnives may also taken as showing what is the piopoition of the constituents in mixed liquid at each stage \AhiIe condensation ol the mixed gas icccds Thus when an containing 21 pei cent of oxygen begins liquefy, the liquid initially loimecl should, nuclei equilibimm idihons, be much nchci in oxjgen the piopoition ol ox\gen il, accojtlmf* to the cui\e is 18 pei cent. These conditions aie appioximateh icahzed when the piocess )\vn as "suiibbmg" is icsoited to m the liquefaction of an. I his piocess, which will be presently descnbed in the toim in uh it Jias been piacticalK tamed out by Claude, a paitial MKihon bclwccn the two constituents is clfected dining the act liqiK. I'.iclion. /ing li^me^ .ue ffivou by Baly Absolute Poicintagc of OM o;cn TompoiahiK 1 In Vapoui In Liquid 77 5 t 78 2 18 8 10 79 80 21 GO 80 1200 33 35 81 1760 43 38 82 23 60 52 17 83 2995 5955 8t 3686 6G20 85 4425 7227 86 52 19 7780 87 0053 8295 88 09 58 87 GO 80 7945 91 98 90 8980 96 15 90 96 100 100 188 THERMODYNAMICS [en. 130 Complete Rectification. In Linde's invention of 1902 the rectifying pjocess is incomplete, for alt ho ugh. the piocess yields neaily pure oxygen it leaves a pait of the oxygen to escape m the waste gas and it does not yield pine mtiogen. In a commer- cial piocess foi the manufactuic of oxygen this is of no consequence foi the ia\v matenal costs nothing, and theniLiogen is not wanted. But a modification of the piocess enables the sepaiation Lo be made substantially complete, should it be desiied to complete it, and allows appioximately pmc mtiogen to be obtained, as well as pure 0x3 gen. The modification consists in extending I he rectifying column upwaids and in supph ing it at the top with a liquid nch in mtiogen. A fiactional method of liquefaction is adopted, which scpaiates the condensed matenal at once into two liquids, one containing much ox^ygen and the othei little except mtiogen The lattei is sent to the top of the icctifying column, while the foimci enleis the column al a lowci point, appiopiiate to the piopoilion it contains of the two constituents PiacLically pine mtiogen passes olf as gas at the top, andpiactically pine cm gen fiom the bottom Fig 03 is a diagiam showing this modified piocess in a lonn gi\ en to it by Claude. The countei-cuiient mtcichangcis which aie ol couise pait of the actual appaiatu-, aic omitted fiom the dicigiiim Compicssed an, cooled bv the mteichangei on its way, uileis llu condensci at A. The condensci consists of two sets ol \cilicril pipes, communicating at the top, whcie they all open into the vessel B, but sepaiated at the bottom. The central pipes, which open fiom the vessel A, aie one set the othei set foini a ling lound I hem and chain into the vessel C Both sets aic nnmeised in a bdlh, .V, of liquid which, when the machine is in full opeial ion, consists ol neaily pine oxygen The condensation of the compicsscd an causes this oxygen to be evapoiatcd Part of it slicanis up ihe icchlymg column D, to be condensed thcic in caiivmg out the woik ot lectincation and consequently to iclum to the vessel below The lest of the evapoiated oxygen, loimmg one of the useful pioduets, goes ofl by the pipe E at I he side In these fcatmcs the appaialus is substantially the same as Linde's, but Iheie is a diflcrenec in I lie mode of condensation of the compressed an Euleimg at J it fiist passes up the ccntial gionp ol condenser pipes, and (he liquid which is fonned in them contains a i datively huge piopoition ol oxygen. This liquid chains back into the vessel A, when it collects, and the gas which has smvivcd condensation in these pipes goes THEORY OF REFRIGERATION 189 110 itf^-L^! ^u,^ ^_____^, n,^ HjLMMTg^g^TBB-iJ Ira D N CHI COMPRESSED AIR J'i Claude's apparatus foi the complete sepaiation oi oxygen and mtiogcn 190 THERMODYNAMICS [e-n.iv on thiough B to the outei set of pipes, is condensed in I IK-MI, and drams into the othei collecting vessel C It consists ahuosl wholly of mtiogen Then the liquid contents of C aie taken (1 1 iron nil an expansion-valve) to the top of the lectifying column, win If those of A entei the column lower clown, at a level L, chosen lo com spoiid with the proportion of the constituents The result is lo secmc piactically complete lectification, and the second proeliu'l of Ihe machine commeicially pure mtiogen passes off a I Hie lop thiough the pipe N and may be collected foi use The action m the central pipes ol the condenser i.s lo be mlei- pieted in the light of Baly's cuives. The first portions of Hie au- to be condensed tuckle down the sides ol Ihesc pipe's and me " scrubbed " by the an as it ascends that is to say they aic bioughl into such intimate contact with the ascending an that ;i condition of equilibrium between hquiel and vapour is al least closely ap- pioximated to. The condition of cqinhbiium when gases of I he composition of an aic being condensed icquiies, as we. have 1 s( t n, that about 18 pei cent, of the hquiel should consist of oxygen 1 . Accordingly the liquid which collects in the vessel A is ol I his degiee of iichness, 01 neai it And by making the conelensei pipes long enough it is cleai that little 01 no oxvgen will be lc-1 1 lo pass over thiough B into the othei pipes It is Irue of ceniisc Ilia! in the uppei paits of the cential pipers the liquid lhal is (01 ined con- sists largely of mtiogen, but as this trickles chw n the pipe- in \\ IIK h it ha^ been condensed theie is a give and lake be I ween if and I IK ascending gas, pieeisely analogous lo Hint which oceuis in .1 i hl\- mg column and when the liquid i caches Ihe bollom il h.is b< c n s> much emiched in oxygen as to be neatly 01 completely in <qmli- bnum \\ith the gaseous an, and theielore ce>nl.uns about IS p< i e< nl. When the 18 pei cent liquid fiom A is disciplined lluoiinh an expansion-^ Ive into the lectilying column al L, il piodm es ,-in atmospheie which has the composition ol air (21 per cuil ol oxygen) Hence the part of the column which e\kncls ,iho\< I Ins point has foi its function to leducc the percentage ol ox^nen m the ascending gas fiom 21 pei cent lo ml, and lln^ is dom^n Ihe second stage of lectification, by means ol the liquid f umi (' \\lneh consists almost wholly of mtiogen |. * That piopoition, as has been pointed out, in spouting of 15,ily'n cm v<--> M lnf< n to experiments made at atmobphouc piessma At Iho hi<.lu,7 pn HSIIKI inuli'i winch condensation takes place m Claude's appuiatus it may not bo o\nc Uy Uin siinic t Forfuithei paiticulais of some of the subjects tmitod jn Ihm (.'hiiplm icloicmo should be made to the authoi'a book on The Mcdiamud Pwdmlwn oj ( '<,!,! CHAPTER V JETS AND TURBINES 31. Theory of Jets. We have now to coubidei the manner Inch u jcl is foimed m Lhc dischaigc, Ihiough an ounce, of steam 1113^ olhei gas undei piessiuc To simplify mattcis it will be mied Lluit Lhc lluicl Lakes in no heat and gi\ CD out no heat to LI bodies dining the opeialion, in othei Avoids thaL the jet is nod nuclei adiatheunal eondiLions Suppose a gas to be flowing -Hiyh a nozzle 01 channel of any ibim, 1'ioni a icgion \\hcie the -.sine ib iclatis el}' high Lo one where it is lowci Each element he slieam expands, and Llie woik ^hich il does in expanding s cncigy of nioLion Lo the iient in fionL of it The \\holc ain Lhoieloic acquni's \doeit\ he piocess and also incieascs olume Lit A and /? (hg (Jl) nuayinaiy pailitions, aeioss c'h il llous, lalv( u a I ugliL angles lu- clneclion of the slieani lines, ic inn in I he ngion of highei sine Lei /* he tlie piessinc '/, v a Ihe \elocil \ Iheu, and V a Ihe volume uLily ol Ihe gas has as it passes I he imagmaiv paililion aL / il.ulv let I'f,, v b and }\ be the picssnu , velocil\, and \ohimcof (pi.mlity at IL Let E a and E b be the mlcinil enei^\ oJ' the <U ^L and B iespeeli\ r t 1> In llo\\mg fiom A lo Ji the \elocily igcs iiom v a lo v t , and Lheie is eonsequenlly a gain ol kmelie gy amounting, j)cr unit of mass, Lo '' t -'- . ^to aeh unit quanlily of gas that enters the space between A and is wnk done upon it by the gas behind, amounting Lo PJ r a lapsing out of this space at U it docts work on the gas in liont lulling to PjfVj, In flowing iiom A to B it loses internal gy amounting to E a E b Hence by the pimciplc of the Fin (,(- nniL 192 THERMODYNAMICS fen conseivation of energy, since by assumption no heat is taken in 01 given out, . 2 _ 7 , 2 * - ^ = E a ~E b + P a V w - P b V b (1). But E a + PJ\ is /, tlic total heal at A, and & + PJ'\ is / 6 , the tobal heat at 7?, and the equation ma\ consequcnlly be wiittcn u lT ~~ U a" _ r r / \ -- ' a -'a" 'ft H """to The gain in kinetic encigj is theicfoic equal to the Joss ol' lolal heat, or what is commonly called the l heal -drop." We aie ti eating E and / as if they weie expiesscd in \voik units when ex- pressed in heat units they have to be multiplied by the mechanical equivalent J The equation applies as between any two places in the How, and taking the piocess as a whole, fiom the initial condition in which the velocitv is Uj and total heat 1^ to the final condition in which the velocity is v z and total heat / 2 we ha\ c -, a _ ~, 2 2g In mairy piaetical cases the initial velocity is zcio 01 negligibly small, and then W 2 ^-'x-', , wheie v is the velocity acquiicd in consequence of the heal -chop This is the fundamental equation fiom which lo calculate the velocity which an expanding fluid acquncs in a jtl, slailing fiom lest. So far theie has been 110 assumption as lo absence of losses tin on gh faction 01 eddy cm rents II we assume, as an ideal ease, that in the foimation ol the jet the lluid is expanding iiudti such conditions that thcie is no conduction of heal lo 01 horn 01 \villun the lluid and also no dissipation ot energy lluoiigh fuel ion 01 eddies, the heat-diop in the equal ion o o V 2 "-Vf_ JL i JL i> 2fl O is thaX which oceiirs in expansion with constanl cnliopy We have alreadyl seen (Art 80) that this heat-drop is equal lo Ihe area \of the ideal indicator diagram (fig G5) lor aehabalie ex- pan&ion %om the initial to the final slate, 01 VdP. . (5). JETS AND TURBINES ]93 This icsiilt might also be infened fiom the fact that, under the sumed conditions, the gas is ang all the work of which it is Gaily capable, as it expands fiom efhst to the second state, in giving nctic energy to its own si i earn legam of kinetic eneigy is, thcie- LC, equal to the aiea of the ideal hcator diagiam. Assume that we may, with sufficient accuracy, expiess the ex- nsion in the ideal indicator diagiam by a foimuLi of the type 7m constant. Then the area of the diagiam, namely l VdP - (P V P V \ - ( 1/1 ~"^ K2) m 1 'iicc when the expanding fluid staits fiom icst, at piessme P, foim a jet, we have v* _ "2g m r = m [ /Pr" 1 Zl l-^J-JP^, an c'(]iialion fiom which to hnd the \elocity t' when NIC s f.illen lo iinv lowci piessiuc P, undci the assumed coiidiln>us of u \vithoul fnction 01 eddies and with no conduction ol hcdl [iialion (f5) is i paiLicuLii case of Equation (1), namely the case icic the expansion is isentiopic and wheie the icLition of piessine volume in isentiopic expansion admits of being cxpicssed by 3 formula PV m = constant [32 Form of the Jet (De Laval's Nozzle) As expansion of i fluid in a jet piocecds the volume and \ elocity both mciease is easy in Iriclionlcss adiabatic flow to calculate both, and in it way to deleimme the proper foim to give to the nozzle 01 iiinel, to make provision foi the increased volume, having icgaid the mcieased velocity. At any stage the aiea of cross-section the channel lequned for each Ib of fluid dischaiged is equal to i volume pei Ib divided b) r the velocity. It is convenient to kon the area of section per unit of mass m the dischaige, and er wards multiply by the numbei of Ibs or kilogrammes. ra T 13 194 THERMODYNAMICS [en. Let M repiesent the dischaige, namely the mass which passes throuoh the nozzle pei second, X the aiea of cross-section of the stream at any part of the nozzle, v the velocity theie, and F the \olume of the fluid there (per unit of mass), then ,. vX , X V On making the calculation foi a gaseous fluid staitmg iiom icst and chschaiged into a icgion of much lower pressuie, it will be found that m the eailiest stages the gam of velocily is iclativcly gieat, but as expansion pioceeds the mcicasc of volume outstnps the mciease of velocity The lesult is that the laLio of volume to velocity at fiist diminishes, pass.cs a minimum value, and then inci eases, and hence the channel to be pioMcled ibi Ihc dischaige, aftei passing a minimum of cross-section, expands m the later stages. The piopei foira for the nozzle, to allow the heat-diop con expending to a laige diop in piessuie to be utih/ed cis fully as possible in gnmg kinetic eneigy to the stieam, is Iheiefoie one m vhich the area of section at fiist contiacts to a nairow neck 01 ''thioat" and aften\aids becomes cnLugcd to an ex I cut lhat is detei mined by the available fall of piessuie It is on this pnnciple that De Laval's "convci<><.ut-di\ agent' 1 nozzle (fig 66) is designed The thioat, 01 smallest section, is ap- proached thiough a moie 01 less lounded entiance which allo\\s the stieam lines to con- veige, and fiom the thioat out- waids to the dischaige end the nozzle expands m any giadual ln k' Wl mannei, geneially m fact as a simple cone, unlil .in aic-a ol su-hon is leached which will coiiespond to the piopei aicn ol discliaigc- foi the final -\olume and \elonty, the values of which depend upon the final piessuie The divei gent tapei fiom the throat oinvaids is made sullieienlly giadual to preseive stieam-line motion as completely <is is piaclic- able, and so avoid the foimation ol eddies which would dissipale the kinetic eneigy of the stieam A vciy shoil lounded enhance to the thioat is sufficient to guaid against eddies in Ihc eonvc igent portion of the stieam, but m the divergent portion a much moic giadual change of section is icquired. The nozzle shown m the figuie was designed foi an initial piessme of 250 pounds per sq inch and a back piessme of about 1] pounds. liy the back r] JETS AND TURBINES 195 wcssuie is meant the pressure in the space into which the fluid s dischaiged. In the design of such a nozzle the purpose is (1) to make the hschaigc have a given value, and (2) to give the stream as hjgh i final velocity as possible by utilizing completely the energy of he fluid m expanding down to the back pressuie The data for the lesign are the initial pressuie, the back picssure, and the intended nnount of the dischaige It Mill be shown as we piocecd that the irea of section at the throat depends only on the initial pressuie aid the intended dischaige, and that the enlaigement fiom the Inoat to the final section depends imlhei on the back pi ess me Against which the sticam it. to escape. At any place m the nozzle the dischaige per unit aiea of crost>- cction is M v ~X = V ' \.t the thioal, whcie the cross-section is least, this is a maximum Consider now the ideal case of isrnliopic expansion in a nozzle \ r hen the fluid is one foi \\hieh PV" 1 is constant dining such ex- lansion Equation (0) is Hum applicable The \docity at any icnnt, the picssmc [hcic ha\mg Jallin to P, is P ,'"-'" V m - ] , p >J 1 ncl Ihe \olume is V = V 1 ( * ]' luice loi tiic disehaii>e pel mill aica oi sccluti; at the place \vheie I a k piessme is P, \\c ha,\ i> /*& \l r 'Ins may be applied to calculate the piopei sec I ion X foi a gn en Lschaine M when the, picssmc has fallen fioin the initial picssmc 'j lo any assigned IOMCI picssmc P Foi the puiposc of designing no/zle there aie only two places wheic this calculation has lo be lade, namely at the Lhioat, and at the end wheic the fluid escapes "cimst the assigned back picssmc. When the throat-section X t nd the final section X f have been calculated, a suitable foim foi the oz/lc is icadily diawn, any smooth curve will sci\ e foi the con- crgcnt entrance, and any conical tapci may be selected foi the ivcigent extension fiom the thioat to the end, provided it is cithci so abrupt as t6 interfere with stieam-lme flow, nor so 132 196 THERMODYNAMICS [en. gradual as to make the nozzle unduly long and thcicby introduce unnecessaiy fuction To calculate the final section X f which will allow Ihc energy of the, fluid to be fully utilized by expansion down to the !is.signcd back piessme, that pressure is to be taken, foi the value of P in Eq (7). To calculate the section at the bliioat the piessme Ihere has fiist to be found The piessinc at the I hi oat is dctei mined by the consideiation that the dischaigc pci unit of section (ftf/X) is theie a maximum. If the expression foi MjX in MCJ (7) is dit'ler- entiated with i expect to P/Pi and the dilfcicnluil wiillen equal lo zeio, the lesultmg value of PJP t Mill be that Tor which MJX is a maximum, in othei woids it will be the value of 1 /*,//-^, where P, is the pressure at the tin oat Eq (7) ma} be wntten __ X~ V m-~l F The condition foi a maximum is found bv d)ffcieiilialinj> Ihe quantity undei the second lool m\P 2 fiom which - P l \ 2 J (in -|- 1 / ^ s )' Fuithei, b> substituting this m Kq ((v/), we have foi UK- \ eloeity at the tin oat ^V-Wi ("J- The volume (pei Ib ) of the fluid at the lhro.il is (10) By combining tlia>e an equation is oblmucd tor Ihe pei unit of ci oss-sec turn at the thioal, __ X t V t ~\m + l) V (n, | 1)T, From this equation the cioss-section al Ihc [1 U()}|1 ls found which will give an assigned discharge when Ihe imlial j)iessuie is known. The ratio of the cioss-socLion uL any place, where the JETS AND TURBINES 197 iicssiue is P, lo the cioss-scclion at the thioaf, is readily found torn Eq. (7a) . //PA- Y v (pj ' A ^ x ^'_ - (12). //Hi V ' P\~^T 'his expression is convenient in dctxi mining the pioper amount f enlargement of the nozzle fiom the tin oat to the end when the aek pi'cssuic ii> assigned. 133. Limitation of the Discharge through an Orifice of Given ize IL follows iiom these equations that the dibchaige thiough given orifice undei a given initial piessuie P x depends only on he ci oss-sec I ion at the nan owes b pait of the oiifice, and is mclepen- ent of the back piessuie, piovidcd the back picsbinc is not greater ban P L as calculated bv Eq (8). By continuing the expansion in divcigcnt nozzle uflei the tin oat is passed, the amount of the ischnige is not inci cased, but the fluid acquues a gieatei \elocitv cfoiciL le.nes the noz/le, because the range of piessme which is Ifective I'oi pioducing velocity is increased To put it in another r av, we may say that the heat-diop down lo the prcssuio at the uoatdctei mines the amount of the discharge, and the remaindci f the hcat-diop, which would be wasted if thcic \\Licnodivugent vtcnsion of Ihc no/./lc, is uhh/cd in Ihe di\ ugtnt poihon to give clchlional vclociU to the escaping si i cam Tins \clocitv is i>i\cn i a dclmilc and iist-lul diuthon, \\hcicas il Ihuc \\cic no di\ oigent vlciision ol the no/'/lc the fluid allci Icaxing Ihc t)7,zlc, would expand laterally, <mcl its parts would n qiiuc velocity in directions such lhal no use could r made of I lie kinclic oncigy so acquucd C'onsidci what Jiapiuns wilh a nozzle such as that PI I' /ig 07, which has no chvcigcnt extension Fluid expanding Iiom a chambci where the picssiue P } into a sjjacc when 1 Ihc pressure is P 2 ssumc the back picssmc P 2 lo be less than P t as ilculatcd by Eq (8) In that case the picssuic m ic jet, wheic it leaves the nozzle, will be P t , and the fmthei i op of picssuic to P 2 will occui thiough scattciing of the icam. The dischaigc is dctei mined by Eq. (11) It is not in- eased by any lowenng of the back piesswe P 2 , because any 198 THERMODYNAMICS [en loweimg of P 2 docs not affect the final piessurc in llic nozzle, which remains equal to P t Osboine Reynolds' 1 explained Ihc appaient anomaly by pointing out that the sticam is then lea vino the nozzle with a velocity equal Lo Lhal with which sound (01 any wave of extension and compic&sion) is piopagated in Ihc lluid, and consequently any i eduction of the pressure P 2 cannot be com- municated back against the stream its effects aie nol felt al any point within the nozzle The picssnie in Ihc sticam at I he oufice theiefore cannot become less, howc's cr low the kick picssmc P 2 may be But if P 2 JCi increased so as to exceed P t , the lalcial scattering close to the orifice ceases, the velocity is i educed, I lie piessure at the orifice then becomes equal to P 2 , the dischaigc is i educed, and its amount is to be calculated by willing P 2 foi P in Eq (7) or (7 a) In applyng these icsults to a nozzle of any foim, the least section is to be legaidcd as the thioat if thcic is a divcigent extension beyond the least section the amount of I he dischaigc is not affected, though the Imal velocity of the stream is mcicastd Taking a, nozzle of anv form, and a constant initial picssiire P, , il' we icduce the back pressure P, fiom a value \\hicli, Lo begin Milh, is just less than P-,, the dischaigc incicascs until P> reaches the a 1 value P l l' J - j . Aftei that, an) r fiutlici icduction ol P, dots in -\- l/ not mciease the dischaigc But the velocity which Ihc (hud aequnes befoie it leaves the nozzle may then be augmented by lowciing P> and adding to the divcigent poition of I lie nox/le The- no///Ie will be lightly designed when it piovidcs foi ]iist enough expansion to make the final piessuic equal lo the back piessme, Ihc |el then escapes as a smooth sticam, and the cncigy of expansion is utilized to the full. If the nozzle does not cany expansion I'ai enough, if in othei woids, the final piessme exceeds I he back pressure, eneigy will be wasted by scalleung. If on I lie olhc i hand the back picssmc is too high for the noz/lc, so llial llic no//lc piovidcs foi moic expansion than can propcilv take ])Iace, vibiti- tions aic set up which cause some waste \ m . \\ r c shall now oonsidei the application of these general rcsulls to an and to si cam 134 Application to Air. In applying the above foimulas lo * Phil May Match, 1SSG, Collected Papa s, vol IT, p Oil f For experiments on tlio effects of no/zlcs M'hich cany oxpansirm ton far, or not fai enough, see Stodola's book on the Steam Tuiljino 7] JETS AND TURBINES 199 my penn anen t gas, such as an, the index m is y, the ratio of he two specific heats (Ait. 25). Its value foi an may be taken as L 10 Substituting this numbei in Eq (8) we have, foi a jet of in expanding under isentiopic conditions, = 528. Hence if the jet is being dchveied against a back piessme less Llum 5'2SP l a diveigent extension of the nozzle is requned to 4ivc the f>icatcst possible velocity to the issuing stream, though Lhe quantity dclucicd will be the same as that which Mould be delivered against a back pressure ol 528P-L If the back piessure be mci eased it must exceed 528P X befoie theie is any diminution in the dischaigc. As a nunieiical example, suppose that air, with an initial piessuie D!' 300 pounds persq inch, is dischaiged thiougha comeigent-clivei- <>ciil nozzle into the atmosphcie, 01 against a back piessuie of sav ]5 pounds per sq inch The piessuie at the thioat is 158 -4 and, since Liu final inlio of piessuics is one to twenty, the uitio of the (mal cioss-scclion to the cioss-scction of the thioat should, by ](' ( , MO\ ] K i 1 V " h X f= VO 101 0-0.3.3 19 =0()() X r Vo 013S5 - 00588 This is 1 01 the ideal case of isentiopic expansion Eftects of Inchon air disicoaulcd, they Mill be consiclciecl m Vit 140 135 Application to Steam In applying the geneial equations loi iscnliopic expansion to steam, \\c ha\ c to distinguish between Hu h pe of expansion uhich occnis in a jet and the type ol expan- sion \\hich was LicuLcd of in Ait 78 In that aiticle the expansion \\as assumed to be isentiopic (adiabatic), (/> was constant But it \\iis also assumed that ill each stage in the expansion the fluid Mas m HUM mal cqiiihbiiuni, it theidbic consisted of a mixtiue of s.iluialed steam Milh the piopoition of \\atei nccessaiy to keep [he entropy constant The expansion dealt with in that aiticle nuiy be ckscubcd as the cquilibiium type of adiabatic 01 isentiopic It is now recognized thai the eqiuhbimm t\ pe of adiabatic ex- pansion docs not occm m Lhc foimation of a steam jet Foi icasoiib which will be appaicnt as we proceed, the steam in the jet is not a mixtuie of saturated vapom and water it is moie or less 200 THERMODYNAMICS [CH. supeisatuiated \\hen the equilibiium condition would be one oi' wetness At any stage of expansion, the steam, instead ol' being in the stable state coi responding to its piessme, is in what is called a ineta&table state, a state that cannot be permanent in any vapour In the metastable state the steam is snpersatuiatcd, it may be completely diy, 01 it may have some watei mixed wilh it, but necessanly less than there would be in a stable mixture at the same piessme In other woids a melaslablc state exists only before the propel fraction of the \apoui has become liquid. In passing fiom the metastable state to the stable 01 equiJibnum state, at the same piessme, pait of the \apoui is condensed, heat is accordingly gnen out, the temperature uses, and the en I ropy of the fluid as a ^hole is mci eased If the steam is supeiheated to begin with, il behaves like a gas in the initial stages of the expansion, and its cquilibiium at each stage is stable until it ciosses the boundai}' or satuiation hue, that is to say, until its temperatuic falls to a \aluc Avhich cone- sponds to satuiation at the piessme then i cached IL is only m fuithei expansion, bejond that stage, that a metastable condition can be pioduced If the steam is initially satmatcd a metastable condition is pioduced as soon as expansion begins According to Callendai's equation, the adiabalic expansion of supeiheated steam follows the kn\ (Ait 78) P(V - &) 13 = const, wheie b is a small teim repiesentmg the volume of walci at C , namelv 016 cub ft per Ib The same foimula continues to apply in expansion beyond the satuiation line piouded no watei condenses out, Ihat is to s\, piovided the metastable condition of supersaluiation is so com- plete that the steam icmains quite di} It also applies, midi-i I he same pioviso, m the expansion of initially satuialcd steam The expenments of C. T R Wilson (ahciirly iclcm-d lo m Ait 79) have shown that m the absence of foicign nuclei, such as dust paiticles, and of nuclei due to ionualion f , ualci'-vapom does not condense when it is suddenly expanded unlil its piessurc is laigely i educed, and then a cloud of small Avatcr-paihdes is obseived Even then, howevei, the conditions aie not those of eqmlibmim, for when the expansion is continued a much denser cloud, composed of many moie paiticles, appears at a latci slagc * Wilson, Phil Trans A, vo] 192, p 403, and vol IDS, p 28') ] JETS AND TURBINES 201 Wilson's experiments weic made by expanding an (01 oilier gas) i tin a ted with Avater- vapoui, but tlie geneial conclusion would no oubL appty if water- vapoui were expanded alone Given plenty of time, a condition of equilibimm \\oulcl be Cached by condensation of pait of the vapoui on the walls of ic containing vessel, but in the veiy lapid expansion which occui> uring the passage of steam through a nozzle, condensation on the mcr sin face of the nozzle can do hltle to^aids bunging it about he effects of sin face condensation aie insignificant Hence in the irher stages of the expansion, as fai as the thioat and for some ay beyond it, steam behaves like the vapoui v m Wilson's evpeii- icnts befoie the clouel of watei -particles appeal eel, it is supei- ihnatccl anel puictically diy This is true of steam that is initially ttuiatcd, anel afo)twn of steam that is initially supeiheated It follows, as Callendai has pointed out 1 , 'that in calculating the schaige of steam thiough a nozzle with a gnen size of thioat, the si/c of thioat requiiccl foi an assigned amount of discharge, ic piopci foimula lo use, in the ideal case of isentropic expansion, that which lefeis lo super^atuiated, as well as supuhcated, earn, namely p ^y _ ^ j = const lie l<im b is lehitivclv so smnll (e\ee[)l al \ci\ ln<h pussuu) ial il ma\ as a mle be ncgloeled, m which case the equations iead> gi\en will appl\ wilh Ihc \aluc 1 3 foi Ihc index m Thus if \\e lake the loimula as PV 1 l = constant (omitting b) id apply il in Equation (S) lo find I he piessine al the tin oat, we \,\ e p ' = 05157 * i I! tic-count wue taken of the Uim b C'allendar| sho\\s lh.it this piession would bec'Oiue P b 1 0-51.57 130 ' i ''] ic- sm.ill lei in chptudmy on b amounts to less than 0001 \\hcn e unlial pussuie is even iis lugh as 200 jiounds pci sq inch it ;>y, therefotv, be omitted m any piaclical calculation of /Vind ^ c> ay lake 515 us Ihc ratio of thioat piessuie lo initial picssme i a .steam ]e-l Tins applies whcthei the si earn is salmated 01 jjcrhealed to begin with, m eithci case the steam is diy when it " "On tho steady How of Htoaiu tluough a noz/lo 01 Uuottlo " Pioc Jnst Mech g , Fob 1915 - IjOC Clt (1 t 202 THERMODYNAMICS [CH. passes the thioat, and it will be supersaturated thcie unless theic has been much initial snpeiheat. Befoic it was iccognized that a jet of initially saturated slcam is necessanly snpcisaLiuatcd when it passes, the LluoaL, it was usual to calculate the thioat pressure by taking, for the index m, a value appropriate to the equihbimm type of acliabalic expansion. The index 1 135 was gencially taken as applicable ! , with the result of making P t = 577 P^ The equihbimm theory, which is cei- tamly inconect, thcicioic made the picssuie in the lliroal Loo high, consequentlj r the calculated dischaigc for a given size of throat was too smalJ Experiments on the flow of steam thiough no/zlcs weie then found to give a dischaige winch was actually gicalci than that which had been calculated foi the ideal case of no friction, although the effect of fnction would be to make the actual dischaige loss than the ideal discharge When, howcvci, account is taken of the fact of supeisatuiation, by using 1 3 as the index, the calculated discharge is biought into haimonv with the icsnlts of experiment. The ie\iscd thcoiy gives a calculated dischaige shghLly greater than the actual discharge, but with no mote difleicnce than can piopeily be ascnbecl to fnchon Using the index 13 m Eq. (11) we lia\ c, loi the dischaige per unit aiea of section at the tin oat, 10 . , M 2 \ a /2 \ 322x13 /P L X t '23; V 23 V F/ - 3 780 k /?i V TV with pounds and leet as units throughout With the units moic commonly cmplo\cd this gives A/ m Ib. pei sec. _ /P l in pounds pci sq inch X t in sq inches 'V F T in cub. ft pei II) ' as a foimula foi calculating the sr/c of throat in a no/zle that is supplied with cithci satuiatcd 01 supcihcatcd steam On the equihbimm theoiy the numciical factor was 03003 instead ol 03155 The collected theoiv makes the chscluu i>e about 5 pei cent gicatei. Aftei passing the throat some condensation no doubt oceuis m the form of a cloud of .small watei-pailiclcs, as in Wilson's cxpcii- mcnts. But the piocess takes time, and the whole time occupied * Compare, for example, the authoi's book on The Slcam Engine and othei Ileal- ti, Edition of 1910, p 214 ] JETS AND TURBINES 203 V the steam in passing through the nozzle is so shoit that it may e doubted uhether the condensation that occms within the nozzle ocs much to icstorc equilibrium 'It is probable that the steam is .ill to a large extent supei satin a ted when it escapes*. As icgards alculalion of I he final aiea of cioss-section no gicat error will c mtiodnccd if A\C considei the formula PF 1 3 to be applicable noughoiiL, and this founula will also give a good approximation \ estimating the final velocity of Lhc jet 136. Comparison of Metastable Expansion with Equili- num Expansion It may help to make this mattci intelligible we compare move full}- the adiabalic expansion ol steam under icli conditions that it is a \\ct mixtmc in a state oJ equilibrium uouiihoitt, M'llh its adiabatic expansion m a melastable state, in 'Inch it icniams completely chy Let steam expand from an Volume F\s, (>S nti.il slaLe icpiescntcd by a (hi> G.s), m \\hich \\e \\ill assume it i bi- di} and sahnated Tlic cnnc m icpiesciils adiabahc c\- aiision ol Ihe h pc licalcd in Chapleis II and III Al t-.uh slayc I Lli.il pun ess M\c Mind is 21 mivliiie, m slal>lc equilibi mm, of ilmalcd vapom audwalei Us \ olmue at an\ picssinc is de-lc i- imed l>> 1hc inclhodcxplamccl in Vi I 7S The cmvc ab icpiesenls lelasl.iblc- adiabalic e\[)ansiou dining which Ihe sLcdm remains mlc div. Us lonn is dele i mined I)\ Ihe equation p () _^)i ' = const | n both cases the expansion is isenliopic and theieloie * Olisi-i uilions ol tho appoaumco of escaping jets mippoiL HUH ccmc-Iiision They io\v lliaL wlitn steam iiuli.illy diy (but not nocoss.mlv supoilicatod) escapes fioni divoi"cml nn//lo in whu'h it lias cvpandod LluouKhaconsidoiablGratio, nopaitjch^s watui bocomo visiblo until tho steam haa tiavcllod some difitanco from tlio idco you lytodola, Zciltithnft dcs Vacuum deuLi,Jier Inrji>/iieu>e, 1913 f Hoio, and on p l!04, the b ot Callondai's equation is written /3, to avoid con ision with the b of tho diagram 204 THERMODYNAMICS [en. But though the eutiopy and the piessiuc aic the -same at b as at c, the fluid is in two veiy dil'leient states. At b it is a homogeneous gas , at c it is a wet mixtuie At c its tempeiatuie is the Icmpciature of satuiation coiiesponding to its piessme theic, at b its temperature is much lower, being detei mined by the equation P (V /3) = KT, 3^ which makes T/P^ = const. The volume is of course less at b. The heat-diop fiom a to c is the theimal equivalcnl of the woik lepresented by the area eacf, and the hcal-diop fiom a Lo b is I he theimal equivalent of the work lepiesented by cctbf, since bolh types of expansion aie adiabatic (see Ait. SO) Ilcncc the heat- drop is less in the metastable expansion, by an amount that is eqimalent to the aiea abc, and the total heat at b is theieforc greatei than the total heat at c by that amount The total heat I of the mixtuie at c, aftci equihbiium expansion, may be found by the method descubed in Ait 89 01 Arl. 1)0. The total heat 7 6 in the metastable fluid at b may bo iound by leckomng the heat-drop from the initial value / Since the volume at any stage in the metastable expansion is m Then, since 7 a - / - A (Aien eabj ) = A\ VdP, ] P>, ID ,P lt 10 ,/' 7 fl - I, = AP a i* (V a -j9) <IP/P" | A? <ll> 1 PI, ' i\ Suppose now that after sudden expansion lo b, <ik>nn (he cm\o' ab, the metastable fluid at b is allowed to become slnblc by pai I i.iJIy condensing undei constant piessure, without any ain or Joss of heat Its temperatuie will use to the saliualion value lor Ihal piessme, it will, theiefoie, come to have the same lemperaliiie as the mixtiue at c, but it will be somewhat dnei, because its total heat lemams equal to I b which, as ^e have seen, is gieatei Uiaii the total heat I c of the mixture at c Its volume will, Ihoicfoie, mciease to a point d, \\hicli is beyond c If we wnte q c foi the cli j ness at c ol steam that has exj)anded m a stable state, 01 state of equihbiium as a whole, fiom a Lo c, and ] JETS AND TURBINES 205 { for the diyiiess at d of steam that has expanded in a metastable ate to b and has subsequently attained equilibiium, by water paratmg out at constant pressuie, without loss or gain of heat, ic diffeiencc of total heats is I d - 1 - L (q d - q c ) ut I d = /(, and / & = I c + A (Aiea abc) [cnce L (q d q c ) = A (Area abc) In attaining cqiulihiium the fluid as a Avhole has gained entiop3 r , >r (j> d is gicatei than (j) b , 01 ^> a) 01 </> c in the equilibiium state, by le amount that woujd convert the equilibiium mixtme at c into ic equilibiium nuxluic aL d. Thus , , _ L (q d - q c ) _ A (Area abc) <Pd ~~ <P& rn rfi * d J- d Jus mcieasc of entropy is not due to any gam of heat, foi no heat as been communicated to the fluid, it is due to the fact that theie as been an nie\ oisible internal change in passing fiom the meta- ablc to the stable stale We may tJunk of the substance as midcigomg a cycle of changes lailmg fiom a let it expand suddenly and adiabatically to b, icn Jet il change fiom b to d at cons I ant pi ess me without taking i 01 gi\mg out heat Then let it be partially condensed, undei msl.int picssme, fiom d to c, duimg this stage a quantit\ of heat msl be gueii out equal to L (q d q c ) Then lei il be skmlv com- icsscd along Ihe equilibiium adiabatic curve fiom t to a. llus impletes tlic c> cJc VVoilc has bcui expended, equal to the aiea 'jc, and a coiicspondmg quantity of heat has been ic,mo\ed Duimg its tiansition (along bd] fiom the mclaslablc to the abJt state, the fluid passes tlnough a state in which its picssuie id \olume are the same as those of the equilibiium mixtme at But its slate in othci icspccts is by no means the same, it is icn a mixtuie ol supeistituiatcd vapour wilh some liquid, not in uiilibiium, its tcmpc'iatuic is lo\vei and its total heat is gieatei It is scaiccly necessaiy to add that the rcmailcs which MCIC made Ail. 75 about the specification of the state of a fluid assumed tat the fluid as a whole was in equilibrium They do not apply metastable or tiansition states As a numerical illustiation of the comparison made in this tide, assume dry satmated steam at 100 pounds pei squaie inch . expand adiabatically to 35 pounds. In the initial state the total 206 THERMODYNAMICS [on. heat is 661 82, the tempeiatuie is 161 28 C , and the' enhopy is 1 6082. Suppose fiist that the expansion occuis without niy \\alei sepaiatmg out We then have Heat-chop l a - I b = 42 02, / & = 019 20 The tempeiatuie falls fiom 164 28 C. Lo 70 2 C The volume at b is 9 962 cubic feet. If, aftei this expansion, the metas table vapour a II. mis eqm- libiium nuclei the constant piessiuc of 35 pounds pc'i si|iuie inch, without gaining or losing heat, it changes into wel sham at a tempeiature of 126 25 C \\ith a diyness q d = 9 123 and .1 volume V d = 11 210 cubic feet Its enliopy inci cases (o 1 01 I-U. Suppose, on the othei hand, that adiabatic expansion lioiu I he initial state at a occuis along the cquihbiium cmvc (. \Ve I hen have ^ = 616 g0j heat _ dlop / fl _ / t = j/5 oa At the end of the expansion the tempeiatuie has fallc-n only In 126 25, and the steam is a wet mixtuie \\ii\\ diyncss f/ t - 'KT77 and volume F = 11 155 The aiea abc, which rcpicscnts work lhat is lost in I lie Insl method of expansion, is cqui\alent to the dilleiencc lu-l \uui llu two heat-diops, namclj 2 40 Iheimal units The loss u hich icsnlls fiom supeisatination is thcitloie ncail\ six pel cuil ol I lie ,\\ a li- able heat-diop This loss of available cucii>v, \\lueli <<cms in a nozzle as a result of supeisatuiation, is distinct horn and addi- tional to any loss that may occm tliioiigh fuel ion 137 Measure of Supersaturation hupcisahnal u>ri m\ol\is supetcoolmg, that is to say, the vapom is cooled In low llx satuiation point coiiespondnig to its pu-ssuie 1 In UK .il>o\( example the supeicooling at b is 50, nanulv 120 2 - 7() y Snpi i- satuiation also involves an excess of picissmc, and a COM < spondmn excess of density, when companson is made bc-hvein llu> |>iessme 01 the density of a supeisatmatcd vapoui and llial ol ,1 .salmahd \apoui at the same temperatuic Thus at b m the ix.iiuple the piessure of the supei satin ated vapoui is ,'35 pounds, wlu-ivas llu piessme of satmated steam at the same tempciatme is only 4 fi/i pounds The latio of densities is neaily the same as I he ratio ol * This is called ' undoicoolmg" by some Miitcjs, but, tlio won! ',.,, ,,,,l,m'" is moie appropriate as a do^nption of cooling wluch w in OVCSH ,,| ll,n amount The expanding vapoui is cooled loo much, not too IiLLlo 1 JETS AND TURBINES 207 L-essures Supersaturated vapour behaves like a gas with PV early constant at constant tempeiatme Either ratio selves as convenient means of specifying the degiee of supei saturation. 138. Retarded Condensation. Wilson found that when clust- ee air, satuiated with watei- vapour at 20 C., was adiabatic- lly expanded by suddenly enlaigmg the vessel in which it was mtained, a cloud of fog paiticles did not form until the volume L' the vessel was nici eased in the latio of 1 375 to 1 This, coire- Donds Lo a ncaily eighL-lold supeisatuiation of the vapour, that to say the ratio of Lhe vapoui densities, 01 of the actual vapoui - lessuie after expansion to the piessuie of satuiated vapour at ic temperature i cached by the expansion, was then neaily 8 [e J'ound that the time-rate of the expansion might be vaned msideiably without affecting this result, and also that when the vpansion was cained fiuthei a much dcnsei log cloud was )imcd, containing man}' moie paiticles It followb fiom these suits Lhdt the gio\\th of those fog particles which MCIC mst umcd did not go on fast enough to icstoie and maintain cqui- biiLim in the expanding fluid Condensation of expanding steam, by the foimation of \\atei aiticles suspended in the Aapom, is uccoiehnglv ictaidcd in t\\o ilfeu'iit wajs Tlieic is uhat ma} be called a static letaidation Inch docs nol depend on the limc-ialc ol expansion, toi the- fog oes not begin to J'oim until the A ohunc has incieased by a definite nd consieleidble amount In addition, Ihcie is <i time-lag which icvenlb cquilibiium horn being i cache el \\lnle 1 tlic expansion jiitmues One icason loi this is lhal Ihc chops, once Lhc^ ha\ c )iincel, must ha\ c tune Lo cool in oielei thai Uic\ ma} continue ^ act as cutties foi condensation Ilcncc the menc lapid the vpansion the lest, neai will be I lie appioach Lo eejuilibiium at any age. altci condensation has begun. It may be emcstioncd \vhcthci, even in such slow expansion as ccurs m stcam-cngincs oJ the ^hnclci and piston U pe% cqui- biiinn of the woikmg fluid is appioxnnatcl}/ attained, noUulh- Laneling Lhe assistance which is gn r cn by condensation on the ictal surfaces. It is quite possible that exhaust steam discharged 3 the condensci ma> consist in pait of supci cooled vapoui lough it also contains water*. Supersatmation in it would be * Compaio Callondai and Nicholson, k On the Law of Condensation oi (Steam," hn PIOL In&t C E vol xxxr, pp 171-174, whoic expenuioiilnl ovidenco ib icnlioncd of uiipoi&atiuation during expansion and exhaust. 208 THERMODYNAMICS [c-u. readuy detected if we could observe the temperature and eompaie that with the piessurc, but attempts lo mcasuie the Umpeialme of supeisatuiatcd steam directly, by means of a Ihennomcli i, oive fallacious lesnlls on account of condensation of -\\alir on Q the bulb, 01 on the pocket m winch it is enclosed, 01 on I he wne if it is an exposed LheimomeUi of the lesislanee lype. The theoiy ol ideal steam-engines using adiabahe expansion, \\lueh was discussed in Chaptei III, and from which ellicicneies ol I he "Rankine Cycle" wcic calculated, assumed a condition of <qui- hbrmm on the paib of the woikuig llmd Ihioughonl I he whole opeiation. So far as there is any departure' Irom thai eondilion m a ical engine iL makes for reduced clheieney in I Ins ,is well .is m othei icspects the real perfomiance ol an engine falls shoil of the standaid set by the llankinc Cycle. In the more lapid expansion which steam undeigots while il passes tluough a tin bine of any lype, U appejus lhal Ihesl.ih is far from being one of equilibrium e\ en in the lalei sl.igrs This view is suppoitcd by an examination ol I he icsulls of I mils ol UK pei foimancc of tin bines, woiking muUi v.uious condilions .is lo exhaust presume and initial superheat '. The icason tvhy diops ol h<pud do not loim h<(l\ enough | ( , pievcnt an expanding v.ipom fiom becoming SIIJK i IK ,iU d \\ill In- dealt with moie full} in Vppcndix I II \\ill In <\pl.im<d HUM that the static ictaidation icfcned lo m Ihis aihele oceuis .is .m effect of suilacc tension in the lupnd 139. Action of Steam m & Nozzle, continued K< now to the action of s team in a nozzle, \\e nuy note in passing howmetastable expansion mny be lepiescntcd on the eiiUopy- tempeiatmc diagiam, 01 on the Mollier diagiam ol cnliopy and total heat. Taking fhsL the enliopy- tempeiatuie diagram (fig. 09), adiabatic expansion fiom an initial state a wheic Lhc pics- sme is P^Lo any lower piessmeP 2 , under e(iuihbnumcomhliojis, is icpiesented by the iscntropic ac, where r, is on the equilibrium hue * See H M Martin, "A Now Thooiy of tho NLoiun 'J'uilund, 1 vol cvi 1918 JETS AND TURBINES 209 >f constant piessme for a wet mixtme at P 2 But if the expansion 5 so sudden as to occui without condensation, it is repiesented by he isenLropic ab, wheie b is a point on the constant-pressuiecuive // for supcisatmatcd vapoui. That curve is a continuation, below he boundaiy line ah, of the consLuit-pressinc cuive for supei- caled steam at P^ TJic ultimate state d which would be reached cquilibimm wcie attained at constant / and constant P (as in g 08, Ait. 130), may be calculated, but it would seive no useful in pose to attempt to lepicscnt on this diagiam the ineveisible ansition fiom state b to state d The diagiam shows clcaily ic amount of supci cooling cb In the /< diagiam (iig. 70) adiabatic expansion under equihbnum jmditions is again icpiescnted by the entropic line ac, the straight line ch 2ing the eqinhbiium line of constant L'essuic I'oi a wet mixtiue at pi Cb sine , Adiabatu' expansion under diy ipeisiitinated conditions is icpie- nted IJN ab, b is ayain a point o\\ ic- constant-piessuie em\e bh ICM pcisalmalcd \.ipom, uhich is a nlinualiou below the boiiiidai}^ i\c ah (A the t'onslanl-piessuie i\e loi siipc-i heated steam al P., .it' \YL m-i\ dettimmc d oiapluc<ill\ ch.i\\ini> <i hoii/onlal stuunliL hue loiinh b lo meet Hie upuhbi mm nsl.mt-piessine line m (/, I he as- umlion being, as bdoie, lli.il the l.istabk vai>our, al'lti expansion, nuaUK eonus lo <i sl.ible slale in d Ihonl ehanuic ol piessuu- and with- t i>ain 01 loss ol heat Tin houzontal <u<>ht hue !>d is a line of constant al ht.aL tn UK so diagiams, as well as in Iig 68, we have assumed that ' si earn is satmaled to begin with But the consti action can piously be modified to apply to steam with initial superheat, 1 point a may be anywhcie in the constant-jnesbuic line for Pj, L^O Effects of Friction The losses that occui in jets 01 bines thiongh friction and thiough snpeisatuiation cannot be 70 210 THERMODYNAMICS [en. sepaiatcd in piadicc, but foi the pmpose of considumg the thei nvxh nannc effect of faction it will bo convenient to Ircat that separately by imagining a case in which thcie is no supoi- satination Such a case may he icahzed by using steam that is highly supeihcated befoie expansion Let BC be the eqnilibaum achabatic curve on the piessuic- \olunie diagiam diawn, say, foi 1 Ib. of steam: then the aica ABCD (fig 71) repie- sents the amount of woik available foi setting the steam D m motion as a jet, or foi C c getting mechanical clfect out ^ Ig 71 of it m any mannei The aiea ABCD is cqimalent to the whole heat-chop in adiabatic expansion nuclei eqnilibaum condilJons, and measmes the utmost woik obtainable in any mclhod ot utilizing the eneigy of the steam It is on this basis thai the woik of the Rankine C}de (Ait 87) is calculated, which loans an ideal stanclaid with which the actual output ot any steam-engine 01 steam turbine ma\ be compaicd The actual output pci Ib ol steam is ncccssaiih less m all cases than the aica ABCD, and the latio ol the actual output to that ana is calleel the "elheieney latio" of the engine 01 tuibme(Ait 91). In a steam jet the output is the kinetic eneigy of the jet itsclt. Considei now the eflect of faction m a ne>7/le Assume the conditions to be adiathcimal If there \\eie no (action (as well ,is no supcisatmalion) the v\hole woik icpiescnted by ABCD woiilel be utilized in giving \elocity to the jet and would appeal m il us kinetic encigy in that case we should luu e V ~ a = Aiea ABCD -- 7 X - /, , to where v is the velocity produced m the jet (slai ling dom icst) and /! / 2 is (in units of woik) Lhe heat-chop m adiabahc expansion But in any ical noz/le thcie is some lad ion between Ihe Ihud and the sides of the channel., and the flow is lo some exlenl tui- bulent, which means that eddies aio formed in which llicic is dissipation of the eneigy of flow Lhiough internal i act ion We shall apply the word faction bioadly Lo all such losses Then elfect is as follows On dibchaige, or at any stage during the ex- pansion, the jet has less kinetic eneigy, and thciciorc less velocity, than it would have at the same stage if there were no Inction. v] JETS AND TURBINES 211 But its volume, after expansion to any given pressme, is gieater Lhan it would be if there weic no fnction, because the eneigy that has been dissipated thiough fuction has taken the foira of heat. Thus up to any stage in the expansion there has been a loss of kinetic cncigy, but theie has also been a gain of heat. Conse- qucntlv the fluid has a gi cater volume than it would have in the absence of fuction Moreover it has a gieater stock of heat still available foi conversion into work in the latei stages of expansion, though that advantage cannot in any event compensate completely foi the loss ol eneigy to which the increased stock of heat is clue The heat that is icstoied at any stage as a lesult ol fuction has losl availability foi conversion into a mechanical foi in, foi the woikmg substance then has a lowei tempeiatiuc than it had in the caihcr slages when the mechanical eneigy was gcneiated out of winch that heat has been pioduced Thus the net icsult is to ml uce Lhe kinetic eneigy of the jet belo\\ the standaul foi no i'liclion, although part of the eneig\ thai has been losl thiough I'nction up to any stage is lecovcicd in subsequent stages The mattci may be put in anolhc'i way b) saying thai, in con- sequence oi fnction, the Iliad, allei expansion to anv piessuie, has snllcied less diop ol total heal than it would ha\ e sufkicd had Ihcie been no liiclion Theie is less mechanical cilecl, but theie is moic IK at kit in it and its \olunic is gicatei, at each stage A j)iogicssivc mcicasc ol cnhop\ occuis dining expansion, as a lesnll of I ho uie\ eisible pioccsscs IhaL aie i>omg ou \\ilhm the ilmd, whcicas with no lnction Hie cnliop^ \\ould I)C cunslanl Taking I he picssin o\ olume diayiam, liy 71, I he el lee I ol Inchon is lo give the actual expansion cmvc a loim such as LtC' , in \\hich the fluid has a gicalei \ oliiino at each slaije than it \\ould ha\ cm tht 1 adiabalic piooiss topic-suited b} UC Bui though this ap- paicnlly implies a gam of woik Iheie is ically a loss The aua AHC'D docs nol mcasine an ac'tual output ol woik, but an ailihcial quanlil} which we ma\ call Ihc "gioss appaicnt woik " Ol this <>ioss appau'iil \\oik, a paiL is icconvci ted into heal, as I he expansion pioceeds, namelv a quantity sufficient to supply enough heal at each stage to bung the expansion cm\c out liom liC to JJC'. At 1 1 ic end ol Lhe opcialion the net amount oi woik thai is obtained, far horn being gieatei than the adiabalic aica AJJCD, is less than that aiea by the equivalent of I z ' I 2 , wheie / 2 ' is the total heat at 6" and / 2 is the total heat at C In other words it is less by the quantity of heat which would be 142 212 THERMODYNAMICS [en required to change the condition of the expanded llmd at constant piessuie from C to C' . To piove Lhis we may think of what happens when the substance is cained through an imaginary cycle. Staiting fiom stale B let it expand, with faction, to C' Let W be the net amount of uoik actually done by it in this expansion. Then let iL be changed from state C' to state C by lemoval of heat nuclei constant piessuie. The quantity of heat so icmoved is /,' - /,. Then let it be com- piessed adiabatically fiom C to B The work W done upon it dm ing the compression (Avhich is reveisiblc) is the same as the C C' G D MM' MM' Fig 72 Fig 73 work that would be done by it in adiabahc expansion The c^elc is now complete, and by the consciyation ol energy we have W + (I/ - /,) - W - o, 01 W = W - (// - I 3 ) Hence also W = I 1 -I Z - (/,' - /,) = /, - //, or the net amount of woik done it, equivalent lo the ai'liwl heal- diop, in agreement with Art. 10-1.. Tmnmg to the entiopy-tempeiatme chagiam (lig 72), the ideal case, without faction, is lepiesentecl by ABCD Friction gives the expansion cuivc some such foim as BC' ', in which the entiopy increases piogiessively as the tempeiatme falls The aiea MBC'M' rcpiesents the heat produced by faction, it is the heat icqmicd to JETS AND TURBINES 213 give the expansion curve its actual form, and since no heat comes from outside sources it is supplied at the expense of the kinetic cncigy of the jet, by a conversion which is going on horn the beginning to the end of the expansion The gioss apparent work is represented 11 by the area DABC', but fiom this. \\c have to deduct the area MBC'M' to lind the net amount of woik which finally appears as kmelic cncigy in the jet Thus allowing for faction the net amount of work ]]" is Aiea DABC' - Aiea MBC'M', 01 Area DABC - Area MCC'M' Hence the net loss, as compaied with the work \\ that would be got in achabalic expansion (with no fncliou), is the area MCC'M', which is /,' 1 }> as f^^ e In lig 7'2 the steam is initially satin ated If it be. supei heated, let B be the initial condition (fig 73), AEB being the con- stanl-picssiiu line foi I\ anel DCC' the const, inl-piessinc line I'oi P, Fi ichonlcss ex- pansion (in cquilibunm) \\unld be H piLsenled b\ BC Die <ie tnal e \pausion is along somi sue h line 1 us IiC' The gioss ,ippai i nl uoi k is H pi e- si nil d \)\ I hi 1 a i ea /) ittBC'd, ,iii(l I hi IK I .unoim! is loiind l>\ di dm ling Iioni I h.il the ~ ,111 .1 M BC' I/', \\ Inch 1 1 pi e v - si ills I hi he <il di"\ i loped I hi o ugh Inchon is lo deducl an amount ol woik equal lo lhe aiea MCGC' M' from I he ulial pel foi manee D itittC This eleduclion is equivalent to // /._, as be I'oi e Foi piacheal pui|)oses il is moie useful to upiescnt the ellccts of Inelion oil the Molliei diagram ol tnliop's and total heal (lig 7 !) Le-l B icpiesent the initial state (in this example theic is some superheat, Ihe 1 bioken line is lire bomulai} eiine) BC repicsents an ideal adiabalic process ol expansion and BC' the actual pioccss /T is the initial total heat, / 2 the total heat thot would be left in Ihe steam alter adiabalic expansion lo P 2 , and /,' is the total heat actually lefl in the steam after expansion to that pressure The * Subject to Uio small corieoliou mentioned in Art 88 Entropy ITi},' 71 I'he net e fleet of diction '21 i THERMODYNAMICS [en. actual heat-diop, to which the net amount of work done is equivalent, is BK or /j - / 2 ', and the net loss resulting fiom fnction is KC 01 I z ' - I 2 When the piopoition is known befoiehand of the friclional Joss KCto the total theoieticallv available hcaL-diop BC we can maik the point K on the achabatic line through B and diaw a homontal line thiough K to find C' . When Lhcic aic ex pen men I a I data i'oi estimating the fnctional losses in expansion clown lo vtuious inlci- mediatepiessmes we can apply this construclion lo fi.ice Lhe actual expansion cuive BC' in a senes of steps The method is applied to compound steam tin bines, as a means ol determining Lhe stale of the steam aftei each of a sencs of stages in the passage oL' steam thiough the tin bine (Ait 1 15) The student may find it useful to cxpicss I he effccl of fuclion thus When theie is no fuction, and the expansion is adiabalie (Ait. SO), (U = y dPj where dl lepiescnts (in units of woik) tlic diop ol tola! heal which takes place while the piessme diops b) dP. When Lluic i^ friction Al > = y d p _ cl Q^ \vheie dl' icpiesents the drop of total heal as affeeled b\ 1 1 id ion, and V the volume as affected by faction, dQ. being UK qiianhlv of heat geneiated by fucLioii at the expense ol Hie gioss ,ipp,iu nl work and restoicd to the fluid as heal lie nee dl- dl 1 = (1Q- (V - V)dP Integiatmg between the limits (i) and (2) wheie Q is the whole quantity of heal geneialed l>y Ii Since 7 X and 1^ aic the same, tins gi\es I z ' -I 2 = q- Aica BCC' of fig 71, which expiesses the fact that in consequence of fncLion I he nel loss of mechanical effect is equal to the heat geneialed, less I he woik that is iccovered thiough the augmcnfalion of volume which fnction bungs about. 141 Application to Turbines The above discussion ol the effects of fnction lelales, in the fiist phicc, to ]els, bul it may icadily be extended to tuibmes When a jet is eonsideicd alone the "efficiencv-iatio" (Ait 94) of the piocess, rcgaidcd as a eon- veibion ol eneigy, is measuied by compaiing the actual kinetic v] JETS AND TURBINES 215 eneigy of the issuing stream with the eneigy obtainable undei ideal conditions, that is to say with the adiabatic heat-drop. But in a turbine, of any type, the piocess of conveision goes fuithei the object is to do mechanical woik on the lotor or revolving pait of the machine, and the efficiency of the piocess ax a whole is measmed by comparing the work done by the fluid on the rotor Avith the acliabalic heat -di op. 142. Simple Turbines In one type of tin bine (De La\ al's) the two opeiations aie enliiely dislmcL The steam enters the tmbine thiough a fixed nozzle of the convcigent-diveigent type (fig. 66) in which Us piessme diops in a single step thiough the \vhole a% tillable range fiom the initial piessure P x to the back picssure P, Subject lo fuctional losses all the eneigy takes a kinetic foim in the ]cL The jel I hen impinges on blades that pioject fiom the cncumfciencc of a vciy lapidl} ie\ol\mg wheel, acting on them just as a jet ol wulci acts on the blades of a Pelton watci-uhcel, \\ith an impulse which is mcasuied b\ the loss of momentum of Lhc sticam The ]el theiebj convcits its kinetic cnfiyy into woik done on the wheel This second comcision is a pinch l^diauhc ])ioccss, a question ol d\nanncs but not of thei moch naniics It imolvcs fiiclional losses (dislinct 1'iom the eaihei hietional losses ID I he no/7le) a^ \\cll as losses aiismg liom the lad that the stieam is not \\holl\ ckpiixccl of momentum In its impact on and passage mil the K-\ ol\ ing blades The eThcii'ii^ -latio of the \\hok piocess is theiel'oic ddeimimcl as the pioilnct ol \.\\o l<idois, naiueh, the clliciuic'\ r -]<ilio of coin eision ol UK steam's hcal-t m iy\ into eiKi<>\ ol motion on Hie pait of UK ]ct, and the ialio m ^Inch lh.it encigv is atlci wauls coiutited into ssoik on UK ioloi II is only m wh.iL aie called ^tm^lc Unbincs that these lactois aie iiulependent, nainth, 1 in bines m ^huh the vdiole expansion occuis in a single step bcloie, the action on the blades begins In some such I ui bines the lotoi has mote than one ung ot moMiig l)hules, and between successive imgs ol moving blades thcic an nngs of hxcd ginclc-bl.ieles which altci the elncction of the stieam but do not contribute any addition of kinetic eneigy, loi thcie is no expansion m them Tuibincs of this kind aie still ' simple" in I he thcrmodynamic sense, for the whole prcssiue-chop is com- plclc belore the momentum of the sticam is utilized, and m all such turbines it is possible to distinguish clcaily between the two lactois. 210 THERMODYNAMICS [cir. 143. Compound Turbines Most tin bines, howevci, aic com- pound, the expansion takes place in a sencs of steps or stages in each of which woik is done on the lotoi Each stage uses only a fiaction of the whole heat-chop, leaving the icmaindci Lo be used in latei stages In each stage thoie is a convcision of pait of Lhc steam's heat-energy into work and theie is fncLional loss both in the nozzle and the blades The heat pioduecd by Lluil loss augments the quantity of total heat which the steam caines on Lo the next stage, theie is, theiefoic, in the subsequent stages a iccoAciy of pait of the loss When the stages aic \ciy nnincions, is they arc foi instance in a Pai son's tuibine, the steps in llu icsnltmg ex- pansion piocess aie so shoit that the pioccss becomes appioxi- mately continuous and may be rcpicsentcd by a continuous curve on the pressuic-A olume dia- giam 01 on othei diagiams A diagram such as fig 75 then exhibits the complete action, the ontei cmvc BC' is a continuous line diawn thiongh points Mhichicpie- D sent the volume of the steam at the bcmnnmo of each slage, and the dilfeicnce bc'h\eui it .ind DO ~ ' the achdbatic cune BC shows how the \olumc is incuasid as a consequence of all the internal losses thai oeeui as the opualmn piocceds. The diagiam diffcis fiom fig 71 onl^ m tins, that tin cun c BC' is now to be undoisfood as including .ill inlciual fnctional losses instead of only nozvlc fur lion \\hal \\,is said m Ait. 110 applies to the efficiency of the luibuu ,is <i wlioh , and so long as no heat is lost by conduction the equation holds good, /-'-/.=-#- \\cnBCC". whcie Q is the heat gcncialed wilhm the tuibine b\ r fluid fuel ion, // is the total heat actually picseut in the steam il its exit I mm the tuibine, and /_, is, as bef'oie, the total heat \\Juch would be found in it aftci adiabatic c\|)ansiou to the same final picssme 144. Theoretical Efficiency-Ratio. Whelhei the stages aic many 01 few, piovidccl no heal escapes to the outside by conduction or by leakage of steam, and provided the kinetic energy of the current of steam is negligible on its exit fiom the I in bine, the actual heat-drop 7 a / 2 ' is all icpiescntcd by woik done upon the v] JETS AND TURBINES 217 rotoi. Let r} t stand for the ratio of the actual heat-drop to the adiabatic heat-drop. J$y this definition, 7 7 ' 7!,-- J 2 ^' ~ / _ 7 2 1 *2 Uiidci the conditions stated above this fi action expresses the cflicicncy-iatio of the turbine as a whole, namely the latio of the work done on the loLoi Lo the woik idc-ally obtainable by adiabatic expansion thiough the same langc The whole adiabatic heat-chop /! / 2 would be conveited into woik only if the tin bine weie revci bible and theicfoie thcimodynamically perfect. Owing to internal jrrcveisibility the heat conveited into work is less, apait from any lobb of heat by conduction to the outside We may call r) t the theoretical efficiency- 1 ) olio It is what the cfncicncy-i atio would be if the whole actual heat-chop I I I 2 ' wcie conveited into woik 145 Action in Successive Stages The action of a com- pound tin bine is most, cloaih shown by using the Mollici diagiam FIR 7<> of cnhopy and Lolal heat lo exhibit what happens in each step. Beginning wilh Ihc initial pu-ssuic, let a senes of consLant-picssinc hues be diawn, p lt p 2 , p s , etc (fig. 7(i), concsponding to the piossincs at whicli tlic .steam cntcis the successive slaves In the lusl sUii>o the picssuic diops fiom p v to p z , m the second staoe fiom p lo pz, and so on In the fust stage, adiabatic expansion fiom p 1 lo pi would be ie])icsented bv a^, and the length of that line would be a mcasiuc of the adiabatic heat-diop, but the actual liCdL-dioj) is Ihc smallci quantity a^ Still ticaliug the action as adiathcjma], a^ is the heat converted into woik while the steam passes thiough the lust stage The condition of the steam at Ihc 218 THERMODYNAMICS [en end of the fhst stage, and beginning of the second, is icpiesented by the point a z , which is found by drawing a line oi' constant total heat thiongh & x to meet the constant-pi essuie curve p i In the second stage adiabatic expansion would give the line r/,,r, The actual heat-chop, ^hich also measuics Lhe woik clone, is a z b i} and the condition of the s Learn as it passes on to the I had stage is icpresented by a A Snnilairy in the third stage the woik clone is 3 &3, the steam passes to the fourth stage in the condition a. v , and so on The diagiam shows the pioce&s, as canicd down Lo I he boundaiy cuive, with steam initially supeiheatcd, it is readily extended into the wet legion In each stage the fiacLion fib/ac measures the latio of the woik done to the adiabalic heat-chop foi that stage The points 1} a,, a s , etc., he on what is called the "cune of condition," a cuive showing what the condition ol Ihc steam would be as it passes fiom stage to stage on Ihc assumption that no heat is lost to the outside The cui\e of condition conse- quently coricsponds to the outci cunc EC' of fig 75 The total \\oik done on ihe lotoi is the sum of the amounts of woik clone in the successive stages, namely 146 Stage Efficiency and Reheat Factor. Taking any stage of a compound tuibine, the latio of Ihe woik clone to the adiabatic heat-chop, in that stage, may be called the stage eUicienev and denoted by 77.,, thus ^ 7?s = ~ac ' The total woik done on the rotoi and if TI^ can be tieated as constant fiom stage to stage, The quantity 'Lac is called by some wuteis Ihe "cmuula,li\c heat- chop' " This quantity is gieatei than the \vhole adiabalic heal- chop between the initial and final picssmcs, /j - / 0) lo an cxlenl that depends upon the stage cflficiene}^ The uilio Sac /,-/, is called the Reheat Facto) The i cheat facloi is relatively high when the stage eHieicncy is low, 01, in other wouls, when Iheie is much Joss thiough irrcveisiblc action vuthm each stage * See Mr W J Goudio's book on Steam Tiubines (Loni;nians, 1917), p 1S)0 r] JETS AND TURBINES 219 Ti eating rj a as constant we have 77, Sac woik done on rotoi ^ s /] / 2 adiabatic heat-diop inclei the conditions postulated, which make the actual heat-drop L measure of the work done on the lotoi. From the equation _ ft t will he seen that in a compound turbine f] t is greater than the ,tage efficiency 77,, since R is greatei than unity. We might have denned the i cheat factoi by lefeience to fig 75 as _ area ABC'D ~ aiea ABCD ' bi in a compound turbine of many stages the cuive of condition s icprescntcd by EC' and the aiea ABC'D, which was called the 'gioss appaient woik" in Ait 140, is the mechanical equivalent of .he ''cumulative hcat-diop" Tide. The woik done on the rotor is 7 S x aiea ABC'D, and is less than the nicn ABC'D, the efficiency- ratio being ^ 77,_\_nica ABC'D f ~~ 147 Real Efficiency-Ratio The foiegoing expicssions m- \olvc Ihe pioviso I hat llicie is no leakage of heal Hut when iheic is leakage of heal, or appicciablc kinetic cncig} in the steam at its e\il hum the luibme, Ihe actual hcat-diop 1^ // includes a quantity icpicsentuig the loss due to Ihcsc causes, in addition to Ihe woik done on the loloi Lei that loss be cxpiessed as a fi.iclion of the adiabalie heal-diop, naiuelv, '''(A-/,) Then 7, -4' -*(/!-/,) is that pail of the actual hcal-diop which is convcitcd into woik on the loloi. Hence allowing ioi this loss, the net or ical efdcicncy-iatio of Ihe tiul)ine becomes i r 1 2 since t] l is, by definition (Ait 144), the lalio of the actual heat-chop I] J 2 ' to the adiabatic heat-diop The amount of woik obtained fiom Ihc steam is therelore 220 THERMODYNAMICS [en. Wilting t-j r foi the leal efficiency-ratio, its i elation to the other quantities is given by the equation In the process of designing a tuibme a value is estimated foi I he- stage efficiency y^, then the cuivc of condition is deduced, which allowsthe icheatfactoi to be found and also the piobablcvolumeand velocity of the steam at each stage In this way data aic obtained foi deteimmmg the toim of the steam passages. Details of the process will be found in books on the steam luibmc 1 . 148 Types of Turbines. An "impulse" tmbinc is one in which the lotoi is cliivcn entiicly by the impulse of a ]d 01 ]cts against blades, which aic attached 1o it In such tin bines the expansion of the steam occms in fixed noz/Ics, or passages which act as nozzles. The tin bines of De La^al, Cinlis, and Zollv 01 Ratean aic examples of the impulse type In DC Ii\ nl's I lie whole expansion takes place in one step, and the extinction ol encigy fiom the jets also takes place in one step Thcie is a single nng of blades, which must have an exfciemely high vclociU il il is to utilize a fairly laige fi action of the kinetic cneigy of the |ets DC Laval's tin bine is used only foi small powcis. Us efficiency is limited by the difficulty of making a wheel that will inn safclv at an enormous speed. The distinguishing chaiactciishr of the Cmlis tuibme is that the kinetic energy of the jets is exliaclod in steps, by making the jets impinge successively on two 01 nioic nngs of moving blades, with fixed guide-blades between to rlcflu-l llu ]ds, as aheady indicated in Ait. 142 This device allows of a mme pcifect conveision of the energy of the ]cls without icqunmg excessive speed on the pait of the i evolving blades. In some Curtis turbines the expansion takes place in a single stage Others aie compound in the thcimodynnnuc sense, the whole expansion is divided into a small numbci ol stages, and the kinetic cneigy acqmied m each stage is extracted by the use of a seiies of two or thiee nngs of mo^ ing blades In the Ratean 01 Zolly type of turbine there aic many stages, each invoh ing a small chop of piessnre and consequently a moderate velocity of ]ct, the jets in each stage give up their eneigy by impinging on a single nng of moving blades Each ring lims m a scpaialc chambci , and * Sec also Batimann on "Recent Steam Tiuhino Piaoiico," Joiini fi^l Klccl Engineeis, vol ^.LVIII, May, 1912, JETS AND TURBINES 221 he jets aie fonnod by nozzles or passages in the chap In agin eparates one chambei fiom the next A "leaction" turbine is one in which the* nozzles 01 passages in vhich the steam expands aie themselves the moving pait, and aie Iriven by the reaction which results fiom the fact that the steam s acquiring momentum as it passes thiough them An ancient oy descubed by Ileio of Alexandria, in which nozzles were caused o revolv e backwards by discharging steam into the air, is an xamplc of a pure reaction turbine. The type has not come into ise, it would rcqurrc an enormous speed of recoil to work efficiently. 3nt a combination of icaction and impulse is applied in the most mpoitant turbine of all, that of Sir Charles Pardons, which was the list to be developed on economic lines, and is moie extensively ised than any other foi generating power on a large scale Parsons' s a compound tuibine with man}" stages. Each stage comprises a ing of fixed blades, projecting inwards from the case and making ip conveigent passages \vhich act as nozzles, and a ring of moving )lades projecting outwards from the rotoi The rings of fixed iladcs> and moving blades altei irate from end to end of the turbine aid aie alike m shape The mo\ ing blades, like the hvcd ones, nake up convergent passages \\hich die completely filled by the .Icani as it passes thiough In each set oi passages, moving as veil as lived, their is some expansion, consequently anv ling ol no\ ing blades is ingcd to move not onlv by the impulse ot the jets vlnch si like it, but by the reaction that arises fiom the expansion )t steam within it, since that expansion gives the steam new idocitv. The nencial chiection in which the sle.un ilo\\ r s thiough lie liubmc is paiallcl to the avis In an oail\ loim ol' I he Paisons luibinc the genual dncetion of low was laclial, the fixed blades being attached to a dvccl disc, and Lite moving blades 1o a paiallel disc which i evolved about ,m avis lluough lire centre of the fixed disc. An mlucsling modification >f this arrangement has been made by Ljungstiom, \\ho lets both "hscs revolve, but in opposite duecLions In the L|imgstioin tiu- binc (which is also compounded of many stages) theic aie, there- fore, no fixed blades, bolh sets arc urged by impulse as mil as by Leaction, and a high lelatne velocity, on which the stage elliciencj' depends, is obtained with a lower frequency of revolution Other turbines arc made up by combining the various types which have been named. It may be added that in compound turbines vvith many stages 222 THERMODYNAMICS [en. the drop of pressme m each stage is so small that the nozzles., 01 blade passages which act like nozzles, aie not of the con vci gent - diveigent kind described in Ait. 132. They aic only convergent, i'oi the drop of piessuie in each stage does not involve expansion beyond the "throat " In each stage the passages must be made sufficiently larger than those of the piecedmg stage to allow foi the mcicasc of volume that has taken place, in the final stages, when the piessme is approaching that of the condenser, the passages arc lelatively veiy laige. 149. Performance of a Steam Turbine. In practice the steam tuibme, especially in large sizes with high initial piessme and high vacuum (that is to say, low pie^suie in Lhe condcnsei) is moic efficient than the piston engine, in the sense Ihal H conveits into effective woik a laiger fiaction of the heal which is supplied to it. For this icason, as well as foi its, gieatci mechanical sim- plicity, it has quickly come to be the duel means oi couveiLmg heat into woik on a laige scale, m pcwei -stations and in the propulsion of ships. As an example of its peifoimance the following figiucs ma), be quoted fiom a tual of a Paisons' tin bine emplo) cd to dine an electiic geneiatoi which developed about 5000 kilowatts The clcc- tncal output Mas measiu eel, along with the amount of t>kam \\lueh passed tin o ugh the tin bine in a gi\ en tune The luibinc iwis found to use 1319 Ibs of steam pei kilowal t-hom of cltctucal output One kilowatt-horn is the equnalent ot liWG thum.il units (pound-degiees ccntigiade) Hence of the heat-end g\ supplied in each Ib of steam r a or 113 7 theimal units wcie converted xo J. J into electucal eneigy If we allow for the loss of powci in the fncLion ol Ix-anngs and in the electucal geneiatoi, by taking the electucal output as 94 pei cent, of the woik done on the lotoi, it follows lli.it each 1137 Ib of steam was doing woik on the lotor equivalent, to - - 01 91 152 9 theimal units The steam was supplied at an initial pressure of 21 ! 7 pounds per square inch (absolute) and was supei heated 07 to 2GI. 7 C The initial total heat was theiefoie 709 1 and the initial entropy was 1 6257 Thecondensei pressme was 47 pounds pei square inch Adiabatic expansion down to that pressure (under equilibrium conditions) JETS AND TURBINES 223 mid piodncc a Avet mixtuie with a total heat of 4840 The mbalic heat-chop Avas theicfore 225 J- theimal units. Hence the uvtio oi' the work actually clone on the lotoi to the 152*9 mbatic heat-chop was ^ r or 68. This is the leal efficiency- Uo ^ ? If one might assume that the heat losses amounted to pci cent of the acliabatic heat-diop, so that x = 06, then the eorel icul elficicncy-iatio -r\ t Avould be f\t = i?i + 06 = 74. i'uiLlioi, the stage efficiency Aveie, say, 07 the coiiespondmg heal factoi would be about 1 06 These mmibeis aie conjec- lal, but tlicy may serve to ilhustiate the meaning of the seveial umlilies, and then geneial oiclei of magnitude 150. Utilization of Low Pressure Steam As Avas buefly unUd oul iti Ait 95, the chief leason Avhy the steam tin bine is a cue efficient means of com citing heat into Avoik than the piston OUR, is its t>iculci power of making effective use of the energ\ low piessmc sLcam In the i eg ion of high pieusuie it has no \, ullage over the piston engine, but in the latei stages of ex- nsion il is a fai bcttei agent of conveihion, for it continues to be leient do\\n Lo Ihc hmcsl piesbiue that is piacticall\ attainaljle .) c'oiuknsei In a piston engine, on the othei hand, it uould be (kss ID cam expansion so fai, foi not onl} \\ould the bulk of ( e\ limit i bceoiucimpiacticablc, but the mcicased \Ad^teof powei i oily li (uelion between the piston and the cUmclei \\ould become ealci I ban life gain ol indicated A\oik Hence, A\ith a piston gine, expansion in the cvlmdci is seldom in piactice cained beyond i absolute pies^me of 7 poundboi CA en 10 pounds pei scaiaie inch ilh a 1 1 11 bine I he expansion is continued ellecti\el) almost >wn ID Ihc condensei piessuie, and it is a mattei of the utmost nsc<nKiiee lo make that as low as the tempciatuit of the con- nsmg \v,ilc r will alloAs Tins poml \\ill be appmcnt if we use the entiopy-tempeiatuie agiam and coni])aic Ihc woik obtainable (undei ideal acliabatic millions) when expansion is complete down to a lou condenser essuu, Avilh Ihc woik obtainable \\hen iclease takes place at a cssmc'of say 10 pounds absolute In the diagram (fig 77) the aiea UCD i-cpiescnts the work obtainable m the complete acliabatic pansion of miUally satin ated steam fiom a piessuie of 130 pounds a condensei prcssiue of 5 pound, and the aiea ABCEF repre- nls Ihc work obtainable when release takes place aftei expansion THERMODYNAMICS [err do\\n to 10 pounds absolute, EF being a line of constant volume (Ait 96) The same condenser piessuie is assumed in both cases The aiea FED icpiescnts what is lost by incomplete expansion, such as necessarily occuis in a piston engine The figure applies to an ideal peifoimance in each ease, with adiabatic expansion, but in the conditions of actual woik the steam turbine would save most of the aiea FED. It is to be noticed thai any i eduction of vacuum will dimmish the output of work J'iom Liu Liu bine much moie than it will dimmish the ouLpuL horn the piston engine, foi when the line AD is raised it afteets the tin bine aiea ^ 2 along the whole length of ,1/Vwhcieas U alfecls the pMon-uM>me aiea ASCEF only along the short distance IF The importance ot high \acimm in a slcam Imbim is besl ivd- ized by woikmg out values ot the adiabat.c hcal-d.oj, w,(h A ...nms back pressuies. Taking initial conditions such ,,s ,,.i found u, ,, tice, with p ieS Muie anywheie between, say, 100 and 200 pounds pei squaie inch, and modeiate supe.heal, the mlil>nl.c hc,,(-dm,> is mcieased about 10 pei cent uhcn the eonden.se, nussmc , s leduced liom 98 to 19 pound, which couesponds lo ., ,,n- piovement of vacuum liom 29 Lo as mchts, w.th the kuometer at 30 inches CHAPTER VI INTERNAL-COMBUSTION ENGINES 151 Internal Combustion. In an internal-combustion engine the fuel which is to supply hcat-eneigy for conveision into work forms pait of the working substance, and its combustion takes place within the vessel or system of vessels in which the woikmg substance does woik by expanding. The woikmg substance, theie- foic, undeigoes a chemical change cluung its opeiation and the bhcimodynamic process is not cyclic In the eaily stages, before combustion, the substance is a mixture of fuel with an, geneially in excess of vdiat is icquued to prov ide enough 0x3 gen toi complete combustion In Lhc Litci stages, aftci combustion, it is a mixtuie >f the pioducts of combustion with, mtiogcn and svith any suiplus >1 air The lucl commonly enteis as a gas 01 v apom diawn in along iViLli a MII table piopoi i ion of air, but it may be injected as a liquid, jecommg vapon/cd aflci admission or ducctly buint on entiy \s a mlc the only chemically active constituents of the futl aie i)du>gui, h vdioc'tiibons and cai borne oxide In then combustion hey mute with oxygen to foim watci-vapoin and caibomc acid riu 1 nitiogcii of Ihe an takes no po.it in the chemical pieces-. K-yond acling as a diluent Typical examples of internal-combustion engines aie the oidmary jas-engme 01 the pcliol motoi, in which a "chaige" of an mixed vith combustible gas 01 vapon/ed liquid fuel is drawn in by the )iston, then composed into a cleaiance space, and theie ignited >y an electiic spaik or other means, so that explosive combustion akcs place while the volume of the chaige is neatly constant. L'he heat I hi is internally developed gives the working substance high tempeiature and piessure it then expands, doing woik as he piston advances. In all modem engines of this class the charge bi ought to a fairly high pressure before being ignited. It will >e shown later that this compression secmcs thermal efficiency, pith inci eased compression a larger fraction of the heat of com- ustion of the fuel is converted into effective work. 226 THERMODYNAMICS |rn From the theimodynamic point of view mlernal-comlmslion engines have this advantage ovci the steam-engine, lha.1 I ben woi king substance '"lakes in" heal (bv iK <>un combnslion) al .1 much Jiighei tcmpeialme In the (oiHl)iislioii of I he chmge n lem- peiatuic of 2000 C 01 so is leaelied The avciagc lempeial me ,il which the heat is developed is far abo\ c thai al which lu'al is icceived by the working subslanee of a sleam-cngine On the- olhci hand iL is not piacticable Lo diseluige heal al neailv so low .1 lowei limil But the aelual working lange ol Icmpeialuit' is so wide that a gas-cnginc can in f.iel eonvei I mlo woik a laigei fiaction of the heat-cncigy of the Inel llian is eonvei led !>N an\ engme which bums its fuel lo laise sleani in .1 boili r, and uses I In- steam, however efficiently, as woi king subslanee \. good gas-engine will conveit about 30 pci ccnl of I he eneigy of ils IIK 1 into uoik the best steam-cngmcs eonvei I no moie llian aboul L() pc-i e nl 152 The Four-Stroke Cycle Tn I he most usual lype <>l in- ternal-combustion engine the mcehinieal < yele is eomph led MI loin stiokes 01 two revolutions Dining I lie In si loi w.nd slmke, gas and air are diawn in, so lhat the whole e> lindei is lill< d \M! h < \|losi\e mixtme, at practically atmosphci ie piessuie Dining llu Insl l.uk- stroke this mixfcme is compiessed mlo .1 eh aianee space al I IK ( nd of the cylmdei The mixluie is Ihen igmUd, \vlnl( llu pislon is ,il or close to the "dead-poml " 01 exheme ol ils ha\ 1 1 The pn SSIIK consequently uses to a much lughei \alne llian w.is icaclird li\- compression. Dining (he second forw.nd slmke Hie liud nu\lui< expands, doing woik and falling in pussme Dining UK s<<oiid backstioke it is dischaiged lluough ,m e\li.iiisl-\ al\ e nilo UK atmospheie A small quanlilv of llu bninl nuxlini K mains m the cleaiance space, and is nu\ed wilh Ihe IK \l (hai'u unless special means aic taken lo lemove il, by whal is e.dNd tl sca\( n- ging." As a rule theie is no scavenging The fom-stioke cycle was ihsl deseubed by lienu de Itoelias in 1862, it was biought mlo use by Olto in IS7(>, and is ollen called by his name. IL is si ill the most usual mode of achon, nol with- standing the practical diawback of having only one woi king si ioke out of fom, a drawback which arises liom Ihe fael Ihal Ihe working cylmdei seives also as inhaler and compiessing pinup 153 The Clerk or Two-Stroke Cycle To escape this d< led of the Otto cycle, Sir Dugald Cleik mtiodueed m 1H81 mi c'ngme wliich completes its action m Uvo strokes Clerk's Engine has a 228 THERMODYNAMICS LCII. in the (iist stroke ol the next cycle. It' the ideal engine were of the two-stioke cycle type, the lines AB and BA would be omitted from the indicator diagiam for the woiking cylmdci, which would Lhen consist simply of the figuie BCDEB. From C to D the whole heat-encigy developed by the combustion ol the chaige goes to heat the woiking substance, since by hypothe- sis none is lost by conduction or lachation to I he walls. The heat of combustion can be calculated when the composition of the chaige is known, 01 may be measmed diiectly by binning a sample of the gas in a calonmctei In all cases one of the pioducts of the com- bustion is watei -vapour, and as any water-vapour foimcd m the cyhndei of an internal-combustion engine remains uncondensed Fig 7S Llnoughoul the action it is pioper to lake, in calculating Hit lual dc\ eloped b} combustion, what us called Ihe "lowci " value, that is to say, a value which docs not include the lalenl heal of Ihe \vatci-vapoiu Between C and D the mixture undcigoes <i chemical change which may 01 may not affect its specific volume I hat is to say, Ihe bin nt products when bi ought to the same piessuie and tempcraluic as the un burnt mixture may not fill exactly Ihe same volume In gcueial the specific volume after combustion is a little less, but with such mix tin es as aie used in gas-engines or petrol-engines the effect of this "chemical contraction," as it is called, on the specific volume is so small as to be ummpoitant With mix tin cs of coal-gas and air it amounts to between two and three per cent, in ordinary vi] INTERNAL-COMBUSTION ENGINES 229 cases With some explosive vapours the specific volume is slightly increased (see Art. 158). The changes being in any case small, it is convenient in considering an ideal engine to ignore them, and to treat the working 1 subs lance as if it weie a gas whose specific volume docs not alter Fmthei, the largest constituent of the un- burnt charge is an, and that of the burnt charge is nitrogen, and the specific heat of nitrogen is,foi equal volumes, the same as that of air. Hence for the purpose of. obtaining a simple standaid with which ical engines may be compared, a practice has sometimes been adopted of tieatmg the woiking substance as if it wcie air., to which between C and D theie is unpaited a definite quantitjr of heat, namely the heat of combustion of the chaige. 155 Air Standard It was on this basis that a Committee of the Institution of Cuil Engmeeis* devised what is known as the "An Standaid" as a meaMiic of the ideal efficiency of an internal- combustion engine Resides assuming (as in Ait. 15-1) foi the puiposcs ol then ideal slandaid (1) No tiansfei of heat between the woiking substance and Lhc metal, (2) Instantaneous complete combustion, and (.'3) No change of specific \ohune, I hey made I he fm Lhci assumption ( I) That Lhc specific he.it might be healed as constant (inde- pendent of tcmpcialme as ucll us pic-ssme) II is nun u'cogm/cd that Lint, last assumption is by no means line c\cn nl an, and is slill moic unLiiic oi the mixed gases in I he cylinder ol a gas-engine- It is known that the specific heat mcicases vuth use oL Icmpciatuie to an extent which gicatly affects the action of the engine This point will be considered lalei but it should be said line that because the specific heal of Lhc woiking gas is much oic'ulei al high (cmpcraluics than at low tcmpcialiues, the u au slandaul," as defined by Ihc Commitlee, is an umeasonably high ciitciion Lo apply to any actual perfoimance. The efficiency of a icjil engine must fall ic<illy short of lhat standaid, not only * Iti'jHnt, of a (Jommillot) cm the Ellicionoy of LntcinaL-Comlmslion Engines, Mm Pioc Just C E vok 1 02 amU03 (1905 and 1900) Tho Rcpoi I gives examples ol oiilculaUoiiH i dating Lo actual and ideal poiioiraancos Kefoionce aliould also be jiuido m this connection lo SH Dngald (Jleik's book on The, Qti^, Pcliol and Oil JfniM, veil T 230 THERMODYNAMICS |cn. because of such moie or less avoidable losses as occur through radiation and conduction of heat to the cvhndei walls, bul because the standaid postulates, on Ihe pail of the woiking substance, fin essential quality that is widely diffei en t Irom (he quality ol thcieiil gases of which it is composed JKxcn if Iheie were no loss of heat, the limit of tempeiatmc which the gases reach aflei explosion must be much lowei than that which would be reached if 111* 1 specific heat weic constant. Ilowevei much Ihe heal losses <iu i minimized, the hypothesis of constant specific heal makes the air standaid an impossible ideal It is nevertheless instructive to study the an standard with constant specific heat as a means of examining some ot the ellecls that follow fiora vaiying the conditions of woiking We may apply it foi instance to show hoA\ the efficiency ol the gas- engine cycle is impioved by increasing the compiession Let T and T^ be the absolute tcmpcia- tures of the charge befoie and aftei com- pression, and let T" 2 and T s be the Icmpcia.- tuies befoie and after expansion Fig 7<) shows the cycle with its stages numbeied to coriespond with these si ilfixcs Wnlc/ for the ratio in which the charge is com- pressed befoie ignition, which is also (he ulio in \\lnch il is afteiwaids expanded dm nig ils woiking shoke Then by Ail '_'<>, since the compiession and expansion aie assumed lo be adiabalic, -. CT S ' r ' fiom which also Tj ~ 7 " -= r <> ( 1 } v ' ^i ~ TI '/', V? ' Hcie y is the ratio of K ;| Ihe specilic heal al conslanl piessme to K a the specific heat at conslanl \ ohmie and is I realcd as a con- stant because the specific lic.ils aie assumed lo be conslanl m Ihe "an c^cle" whose efficiency we aie now finding The heat supplied, namely the heal flcnoiMlnf in I he explosion ,s K (T, - T,) The heat rejected is K,, (T, 7 1 ,,), for il makes iio chffei encewhethei the products of comlmslioiune cooled on release to the atmosphere, or kept in the cvhnde. and cooled there to atmosphenc temperaluic.aL conslanl volume, before being released vi] INTERNAL-COMBUSTION ENGINES 231 Hence the thermal equivalent of the woik done in the air cycle is j IT T } K (T T \ and the "air-standard" efficiency is K r (T 2 - T : }~K r (T. - ?') 1\ - T K a (T,-'J\) m } ~ f'^TS /I \ yl which is equal to ~ This c'xinession is nnpoil.inl as showino the beneficial influence of compiession. Much of llie pracLical iinprovcmenL of oas-cnQiaes has in fad lesultcd 1'ioni pio<)icssi\ r cly mcKMsuiq Ihc cxlent to which the chaie is coniiJiessed bel'oie i^inlion WjLh incieascd lalios of compicssion the "an-sLandaid" effi- ciency ineuases as follows, lakinn y to be 1 I. Jlalio of Vii-StniuLinl n lillic ii'iK y 2 U212 { !, r )() I I) 12(} r> o 1.7^ ( Of) 11 7 Of) II S I) r )() r ) 10 OM)J II will be seen lioiu Ihcsc II^IIKS .ind lioiu llu' cui\c (|IL> ( SO) I luil lluie is al liisL a \ ei v i.ipid ^><IIH ol elluieneN \\ill) mcie.ised compiission, buL IhaL ^\llLu UK compulsion is lu^li Ihe llieimo- d} namie advanLi^e of iiH-icasiiio il boconu^ sli^hl \Vheu accounL is Likeu of vaii.ilion m specific heal, liyuies aie obtained foi the Lheort Ueal (.Ilicunev which fall slioil of Lhe an sL.mdaid by about 20 pei eenl (see Ail 107), buL Ihc proporhon between Ihe elli- jiencies for dd'fercnl amounts of compiession is nol ^>ieaLly alLeicd. The ellieiencies aelually obLmicd m trials of engines are of comse considerably lower, owing to heal losses and to the fact UuiL tlie combustion of the chaise is not instantaneous. In veiy favourable cases the measured thcmial efficiency is as hiyh as 37, coircspon- -hng Lo about 08 per cent, of the an staudaid, 01 to about 83 per jent of the theoretical standard that is obtained when account is .iikcn of variations m specific heat This is for engines of the 232 THERMODYNAMICS [C'H oidmary tj'pe in which combustion occurs, al nppioxunalcly con- stant volume, aftei the compression ol a mixed charge In all such engines theie is a pi action I luml lo I lie amount of compiession' it must not be so great as to cause prc-iimilioii by unduly laismg the tempeiatme befoie the end ol the compression stroke This limit diffeis with different kinds of 1'uel, it is com- paiatively low when theie is much hydrogen. In engines using ordinal y coal-gas the latio of compression is in piachec as hii>li <is 6 01 7, m petrol-engines it is usually about '1 01 less We shall sec m the next aiticle that by departing from Lhc constant -volume I yju* ol combustion, higher ratios of compiession become praeheablc, willi some mciease m theoretical thermal efficiency '20 u 2 4 6 8 10 12 14 Ratio of compicssion. r Fig 80 Eflicioncy of " air Simula id " 156. Constant-Pressure Type Besides Ihe consUnl-N chn.u type of inteinal-combustion engine, to wlueh oidiniiiy g;is-c nu m < s and most oil and petiol-engmes n])piox,nui[cly conlin.n, \\^ni,n imagine a type in which the piessuu- of Ihe workinu subslancc does not change while combustion is lakmg place Si.p,, ( ,s ( (h-,1 the an is sepaiately compiessed mlo the cle.uuncc- space belo, e a 1 1 v fuel is admitted and that fuel is then fenced m, bununo- as ,1 nuV, s while the piston begins its foiwaid movemenl Jiy suKably ,e<M,- latmg the late of admission of the fuel Ihe ]),essu,e may be kri,l constant till the combustion is completed. In this imaginary cycle the heat is supplied at eonslant p.cssmv vr INTERNAL-COMBUSTION ENGINES 233 We may furihci imagine the icjection of heat to occur at constant piessuie, if we suppose that bhe products of combustion aie ex- panded adiabatically do^n to almosphciic pressme befoie they aie discharged. The ideal indicator diagiam would then take the form sketched in fig HI. Undei these conditions (which are not leahzed in practice) AVG should have an engine of constant-pi essme type, i ejecting as well as receiving heal at constant piessure Its au- jtandaid efficiency is icadily cxpiessed in a foim coi responding o IhaL found for an engine of constant-volume type We aie once mod heic with Hie specific heat at constant pressme, K v F:^ SI Coii-il mt purlin. 1 h pi 1 'icaling it MS consUml, the heat taken m is K p (T, T t ), the eat rqeclecl is K (T A - T ), and the efliciencv is A- / rit //' \ /' / ( 7 2 J - J l) ~ K I> 'he lalio oCadiabahc expansion is equal to I he latio ? of adiabatic nuiwession, 2\,/7\ = T/l\ Hence the aii-standciid efficiency is iven by the same expies.sion as befoic' 1 , namdv 1 - (T w * It is mleiosling to noto tliat tins same oxpiession applies to three ideal types of igmo (1) Tho constant-volume typo, m which Iioat is ifceived and rejected only at constant volume 234 THERMODYNAMICS fc-ir It follows that foi equal ratios of compression thtvre would ho no thermodynamic advantage in substituting a constant -pressure type of engine for the constant -volume tvpo Hul hy avoiding any admixtuie of the fuel with the an before eonipiession il heroines piacticable to use a higher ratio of conipiossion, and consequently to obtain a highei efficiency 157 Diesel Engine This advantage, is in part secured in the Diesel Engine., which compresses the air sepaiatelv (o a |)iossiiro of 500 pounds per squaie inch or nioie, bc/oic (he I'uel is admitted The an is compressed by the baclcwaid slioke of (he pislon Tin- fuel is oil, which is delivered by a separate pump into I lie highly compiessed an while the piston begins its foiward slioke The oil at once ignites, because of the high temporal me lo which I he ;m has been biought by compulsion Us combustion keeps I IK pies- suieneaily constant until the supply of oil is cut oil 1 Tin pioducls then expand, but not to the extent shown in tin iin,igin,n\ engine of fig 81, for expansion is continued only to u \olnme |iul lo th.il of the an befoie compiession, so that when lelo.ise Likes place, the piessuie is much highei than that ol'lhe .ilmospheic As leg.uds leeeption of heat the action of the Dnsil engine .ippio\ini,iles (o the constant-pi essine type, hut as ugjnds lojcchon ol lu.il i[ approximates to the constant-volume type, and in (hat i< sp I i( s theoietical efficiency is somewhal uduced The high iiiih.il eom- piessionenahlcs it, howcvci, toeonveil moicol I he I luiui.il em IH\ of the fuel into indicdled woik than is eon\ oiled in olhei mh-iii.Tl- combustion engines The piaelieal ad\.inl.ig< ( ,f || lls ls j,, S(MU( extent countei balanced hy its gi-e.il ei mcch.uuc.il lueli m , ulm-h brings the net output of effective woik down toaliguie moie n()1I | N compaiable with that of olhei engines In a tnal ot a Diesel cngmt indicating nenilv KM) huiM'-pmu i l! a theimal efficiency (with legaid (, mdicalcd poue,) ,,| o 17 , s claimed, with a mechanical elhe.enoy ol () 7<> II ||,,-si. lig,,,, s .,ie conect the engine was eonvei tmg :j<j pe, cent . ol I he I he, nul euei ,r v of the fuel into effcetn c woi-k, available for d, u ,, ol he, machin^ (2) The constant piowuio L yjlU) m w |,u h lu, ll( , , ( ,uv,,l and j, , (l ,1 ..... v llL consLaiit picasiuo (3) Tho constant LompouituuUy^ {('aiii|,' H .. MK n. nf li { r I, A.I ,I(,J in uhi< h heat is tooivod^uul .ojoUod only al, (i..n H liuil lnm|>i-iu(iu<. Kui Kn efflcianoy I - , a ,l ' 1IH1 , , , (l( , , ||U||1 (|f adiabalu. compression (nut isotliouual, an in Ai 1, Quoted by Mr Mathol, / OM , l )M n fllll K ll .\h vil INTERNAL-COMBUSTION ENGINES 235 158. Combustion of Gases. Molecular Weights and Volumes In calculations that iclntc to the combustion of gases the quantities involved aie most conveniently reckoned per unit of volume, at a slanchiid condition as to tcmpciatine and pressuic A chief reason foi this is that the densities of oases are piopoi tional to then combining weights, and consequently the volumes m \vhich I hey nmle have a very simple ratio. The combining weights Poi the substances with which we aie now concerned aie (in lound nuinbeis "), Ilydiogcn II = 1 Oxygen 010 Nitiogen N = ]-J- Cnibon C =12 Ilydiogen, ox) r gen and nitiogen arc diatomic gases, that ib to say their molecules, II,,, 2 , N,, each eompnsc two aloins, and then molcculm \\ughK aie accoidmgly 2, 32 and 28 icspecUvcly The volumes represented by these weights aie the same for all Llncc, when biought to the same picssiiic and teni[)ei<itine The equa tion 21 1, + O , - 2 1 F ,O means that in the combustion ol hydiogen two molecules of hydio- gen unite Milh one molecule of oxygen to fonn I wo molecules ol watei As legaids weights, it menus lhal I pails by \\eigl\l ol h>diogcn unite with 32 pai Is bv weigh! ol o\\ gc n to foim 30 |>ai Is by weighl of walci, and lhal I he moleculai \\ciylil of water is IS As legal els \olumes, it means lhal I wo xoliiuus <>l h\diogen unilc wilh one \olume ol o\vgtu lo foim two \olinnes of walei-\i[)om, assuming lhat the compai isou of \ olunus is made nuclei such con- ditions of U mpcialme and piessme that I he waUi-vapoui may be ticalecl as pei feel ly gaseous Again, the cqualion 2CO -h O, = aC'O., means lhal two molecules ol cai borne oxide (SO pails by weight) mule wilh one molecule ol o\ygc n (32 pai Is bv w< ighl) to Joi m Iwo molecules of eaibonie 1 acid (SH )>ails by weighl) II also means that two volumes of eaibonie oxide mule wilh one volume ol oxygen lo lonn Iwo volumes of eaibonie ticid. In Ihe combination of any gases Hie pioj)oi tion by volume is given clnoetly by I he iclative munbei of molecules The pimciplc involved known as Avogaeho's Law is lhat equal volumes of all * More exactly, taking as 16, II is 1 008 and N is H 01 23(J THERMODYNAMICS [en gases (in tlie peifcctly gaseous state and undei the same condition*- as to piessure and tempeiatme) contain the same uumbci of mole eules The weight contained in unit volume- in olhci words the density is theiefoie proporLional to the moleculai weight A feu further illustiations may be useful Weights Volumes Weights Volumes Weights Volumes Weights Volumes Marsh gas (Methane) CIIj CH 4 + 20 2 = C0 2 + 211,0 16 + 01 = 1-1 + 36 1 + 2 foim 1 + 2 2II 2 O ELhvIenc CJIj C,Hj+ 30 2 = 2CO> 28 + 96 - SS -I- 36 1 + 3 form 2 + 2 Butylene C 4 II 8 c 4 H a + 60, = jco 2 + 4n 2 o 56 + 192 = 176 -|- 72 146 foim 1 4- -J< Alcohol C,II 0. C JI^O H- 30, == 2C0 2 + 311,0 JO + 96 = 88 H- 51 1 + 3 form 2 + 3 It will be obseived thai with alcohol and \\ilh llu hcav} hydio- caibons, of which C 4 II S is one, the specific \olnmc is mei cased by combustion, wiLh maish gas and clhylcnc il undeigoes no change , and Avith h^ydiogen and caibouie oxide iL is i educed The clhinne of specific volume which any given gas nnxLuie will muh'igo on complete combustion is leachlv piechcLcd by a]^|)l} ing I Ins method of calculation to each of Lhe constituents oJ the fuel, when Ihc composition of the mixtuie is kuoun AnoLhcr obvious ji|)[)hcalion is to calculate the volume of oxygen, and by mfeicnce the volume of an, requned foi the complete combustion ol a given gaseous fuel For the puipose of such calculations, cliy air may be laken as a mixture of 20 9 pei cent, by \ olnmc of oxygen wilh 79 1 pci ei'iil by volume of mtiogen The folloAvmg example will seivc to show how I he an required for the complete combustion of a gas of known composition js calculated, and also the change of specific volume, 01 the "chemical [J INTERNAL-COMBUSTION ENGINES 237 mtiaclton," winch will take place on combustion The fuel is >al-gas 5 of the composition shoAvn in the fiist column. Vol ot oxygen Composition of lequiredfor the gas < ompleto Volume of pioducfcs ly volume combustion H 2 CO, N, H 2 422 211 422 ('[! 3; o 680 08 34 G n H 4 48 144 96 96 CIH 8 CIO 2 1 (i4 126 32 84 84 JSL 08 OS c'Oj 37 37 1000 1193 128 2 62 1 08 210~3 Hvfl ic chemical contiaction is 219 3 197 1 = 22 2 volumes. Since 119 ,'3 volumes of oxygen aie leqiuiecl foi complete i , M i 110;J " 10 e ~i i mbuslion, I lie an lequued is t 01 571 volumes. wW J I'ncc il llns gas is c \ploclccl in the nchcst possible mivtine, with smpliis an noi olhoi chliicnl, the contiaction amounts to 22 2 a lolal \olnmc ot 071, 01 33 pci cent In a gas-engine \vheie ( mixluit 1 is diluted bv excess an, and bv lesidual pioducts >ni a pic'Mons cliaiqe, the contiaction \\ill of coin se be a smallei ipculion of the whole volume 159 The Gramme-Molecule or Mol The essence of Avoyaclio's [iifiplc may be- put in anolliei \vay by sayni" Lliat if we take mills (i>i, mimes 01 I!)b ) of diffoicnl oases, ni bcmy the numbei ucli K'picsents the mokcnlai weight, I heir volumes (at standard jssiiu 1 and leinpeiriLuic) will be Lhc ^ame- 1 Say, foi example, il Liu 1 chosen unit of mass is Lhc giamme take 32 giammes of ygen, 28 gianuncs of mLiogen, 2 giammes of hychogen and so on, j volume oi each quantity will be the same Ic This quantity, which hi lei on I foi diffeient gases, but has neaily the same volume foi , is called a "giamnic-moleculc" ot "mol " iLs volume is 22,^00 )ie ecntnncLres for Lhe nearly perfect gases, at C. and one atmo- icie f . Pioj)eitics of gases such as the specific heats, or the internal , or the heat ol combustion, aie often stated per "mol." Subject to small differences which aie due to the fact that the gases are not jtly "perfect " Calculated from the density of oxygen 238 THERMODYNAMICS [en- A quantity of heat stated m giamme-caloiies per mol may be con- verted into foot-pounds pei cubic foot by multiplying by 002205 x 1-100 or 3 90, (0 082808 px 22,-! 00 since 1 gi.-caloiy = 002205 Ib -caloiy, 1 Ib.-caloiy = 1 MM) foot- pounds, and 1 cm. = 032808 ft. 160. The Universal Gas-Constant The i>,is equal ion PV = RT, is strictly applicable only to ideal oases winch aic ''pi il'ecL" in I ho sense of oiling Boyle's Law and also Joule's Law (Ail 10), 7' beino the absolute tempeiatuie on the theimodynanne scale It is appioximately tiue of all gases at low or moderate press mes, pio- vided the conditions as to piessuie and tcmpciahuc arc nol such that the gas approaches liquefaction. At any given leinpeialiiu- a leal gas is moie and moie neaily "peifecL" the more the picssiiu is icduced. Wilting the equation in the fonn RT V p ' and multiplying both sides by m, the nunibci which cxpiessts I he molecular weight, we have rr mKT mV= p Heie mV is the volume of ? units ol mass Bui thai volume, as Mi- saw in Ait 159, is veiy nearly the same foi all gases nuclei I lie same conditions of piessuie and tempciahnc Hence in/i is also \eiy neaily the same foi all gases The qnanlily in It is called (lit nm- veisal gas-constant Like R (Ail 18) it is a qnanlily ol \voiL, lobe expiessed in work units 01 equivalent heal umls Us numeiieal \alue depends on the unit of mass that is used m the leelvoning. Let the unit of mass be thegiamme, IhcnwFrcpifsuils I he volume of a giamme-molecule or mol, which is 22,400 cubic centnnehes when the teinperatuie is 0C. and the piessuie is one atmos|)heu (equal, by Ait. 12, to 1032 7 giammes per sq cm , or, m absolute measure to 1 0133 x 10 dynes per sq cm ) Hence the universal gas-constant D 1032 7 x 22,400 = 273 1 - = 8I '' 700 -ij INTERNAL-COMBUSTION ENGINES 239 -Ve mav also express the gas-constant in hcaL units Since, by Ait. 4, the giamme-calory is equivalent Lo -126 7 giamme-mctres 01 '2,670 giammc-cenlimetrcs, ni If - l =1 085 giammc-caloiies* 12, (5/0 Igam, il Ihe mill of mass bo I ho 11) , Lhe "as-constant cxpiesscd in b-caloncs is 1 085 | s which is (qui\alcn(- to ] 985 x 1400 = 2779 DO I -pounds The gas-conslani may be inlcjpielecl as Ihc woik lhat is done y expansion when in unils of a gas aic healed nuclei oonsLant lessmc lluough one degree Knowing the ga.s-consla.nl we can icmlil}' caloulaLc Ihe value of J 111 the equation PT -- KT (01 an\ gas Lo which llial equation pplics, bv di\ ulmg Ihe constant b> in Values of If calculated in us mannei aie gi\ui below ('iili'iiliilf (I valuos df Jf 111 <ri UIIIIK I Illlllli i |)l I <r|,lim>l(i 111 llHll |)(IHlllK HI II) ( ll(>l II 1 |)l I 111 [)l I II) (Kvgta ()()_>( I 8(>S Nido^LMi 0070') )') J Au (MKiSS ')(. { Cuboim <)\itlc 0070') <)!> ,] II should be icealltd lh.il Ihe \alue of If is equal lo Iht dillti- ice Ix'hvcon Ihe specific lu.ils ,il conslanl pitssme and ,il eon- anl \olnme, I\ and A',, (Ait. 20) 161 Specific Heats of Gases in Relation to their Mole- ilar Weights Volumetric Specific Heats In Ail 20 we 11 UHIIHJ iiliMoluit (r <r H ) uiuls, (he slanclfinl iilmosphuiH as doJmod in Ail U in H'$.J > JO" d\ IK H J)( 1 H([ (111 JIlllUO UlO gllS COIlHltinl 1 01. 'W % 10" % 22,400 21, \ J. ~ L ^ o guiinmo-caloiy UH dofinod in Ail 10 is oi^uivalonl to 4 1808 A 10 7 oigH honco S3 n , 10" ? " A>= -4]H(>HxU,^ ]08 ^ UttmmOCalOU08 ' in tho text Tho numoi ical value la uol allot od, 1 Ib calory por Ib being equal lo I gramme- ory por giammo Taking m for Iho mixluro of nikogon and oxj^gon as 28 86, namely 79_1 x_28 20 9 x 32 100 "' " 100 240 THERMODYNAMICS [<'"- reckoned the specific heats K p and K per utul of mass Foi many pmposes it is moie convenient to icckon the specific heats of jjasi-s per unit of volume when so icckoned they aie sometimes called volumetric specific heats Most eoiivemenl of all is lo icekoii I hem per gramme-molecule or mol. This ism olTeeUi volumelne method, foi the volume of the mol is Lhc same m all t>as( s lh.il salisl'v tin equation PV - RT When the specific' heals of such gases are reckoned pei mol their diffeience, is C([iial to the gas-constant. TIlUS K, = K v -[ 1 1)85, when K v and K v aic reckoned in gramme-calories per mol It follows that in all such gases the mho y of A",, lo A',, is 1 OS 5 y = I+ K, The vohimetiic method of icckonmi> specific h< il h.is ( Ins hn lliei advantage that when so reckoned I he spirilu lu.il (A' ; , 01 A,) of the -simplei ga^es is neaiJ^ the same, [iiovidcd the ij.iscs h,i\< the same nuinbei of atoms in the molecule Ml the y.isis ii.nncd m the list in Ait 1GO aie diatomic, .null he\ h.ncm.iiU Hit s.nuc specific heat ^vhen that is leckoucd [>ci unit ol volnnu , 01 p< i mnl This is found to be tiue when the spc'cilu heals of lh<-s< g.isc s an expemnentally measiucd and comp.ucd, bill il c.ui also hi- inl< IK<! fiom the kinetic thcoiy of gases (See A|)[)i i ndi\ II ) The kinetic theoiy shows thai m an ulial 1 diatomic ^as Ky^^R. Theicfoie m any such ^<is K lt ]lf and y ', I 10 This is found to agree "\\cll with the valnesol ygol hydneel measnie- ment in an, oxygen, nitiogcn and olhei diatomic pei nuiin nl njises It follows also that the values of AT and A" (l deduced I nun I he I hi 01 \ , when expiest.cd m calones pei mol, aie foi all such ,ases , r 7x1 !)S5 , T , 5 x ] 085 and 7v = - - = I, <)(i;j From these figuies the following values of the specilie In-als aie * Ideal in tlie sense that the gas satisfies tlio oij[uatiou PV ItT nn.l ulxo |,1 1 at its molecules have no sensible onoigy of vilnation (Ait 17)) vi J INTERNAL-COMBUSTION ENGINES 211 deduced for vanous diatomic gases, by dividing by the value of m nppiopiialc to each. (J lie. ul ik-d ipO( ilic licats m oi lb ' iloncs pei lb A',, K,, Oxygen 02171 IB/51 NiUogi-n 02481 01772 An 02408 01720 ,H7l 2481 c oxido 02181 01772 The obseived specific heals dil'lei a hi lie Irom Ihese, because he leases aic not ideal 162 Summary of Methods of expressing the Specific ieats A shot t summai v of ineLhods of slal my K Jt and AT, in yases miy help lo a\oid contusion Kilhei of Hiese quantities mav be taLed as lollo\\s (a) In oi.unnu-Ctiloiics pei i^iamme-moh enle 01 mol, I lie .itiinme-moleenle 01 mol beiny <i mass equal lo in itammes, uh< ic '/ is the niimbei which e\pi esses the- moleculai \vei<>hl (b) In oiammc-caloiics pci- ntannne (c) In lb -caloi ics pci lb (d) In root-pounds pei cubic loot (r) In lool-pounds pei lb To com ci I 1 1 oni (ti) lo (h) 01 lo (< ) di\ idc b\ /// The mini be is in '>) and in (0 .ue the s.iim To con\eii horn (ft) lo (d) nuilliplv \ r .'5 ')() To con\( il I'lom (< } Lo (c) imilhpK by I 100 The dilhu ncc behvceii A ; , and A',,, \\lucli is neaily constant in 11 ^ascs, h.is llu lollou'inii \alncs fu (<i], 1 ( )S."i caloiies I !).S5 In (b) and (< ), caloius in In (f/), T 71 l(K)l-})ounds '277!) In (c), " iool-|)ouncls m 163 Measured Values of Specific Heats 1 1 is lo be expected r ial the actual specific heats of gases should shghllv exceed tlic alues calculated fiom the kinetic theory,, owing to the depart me i ical gases Irom the ideal conditions assumed m the theory Measurements of K } , by Rcgnault foi a number of gases gave nines which are somewhat less than the theoietical values, but 213 THERMODYNAMICS |c-u. the method used by him is now believed U> have been al'lielcd l>\ a systematic error, the effect of winch WHS [<> m.'ike Ihi ineasined values too small, apparently by aboul "2 per mil * A modem measincmcnt of K for an by Swami |, by means <>[' electnc heating undei conslanLpussine, gives (when i educed lo I hi mean calory used in tins book) A',, -= 02M8 ealones pei uiul of mass, at C and OIK atmosphere, which is, us \ve should e\peel, slightly gieatei than Lhc theoretical nnmbei The conesponding value of K Avould be 17U5, hiking H lo be O 0688 as m Ait. 160 f v has been dnectly measured by Joly for seveinl gases, by the device ot applying steam externally lo hi a I a coppci globe containing the gas, and compaiuig Lhc amoiml of sleam Iheirbv condensed on the suifacc with I he amounl oondcnsid on aiiolhei exactly similai but empty globe). His observed \nlne ol A",, loi air, under stanclaid conditions, I hat is lo say al 0" (' and one .1 lino- sphere when coriected for Ihe icviscd \alue ol Hie l.ilenl beal ol steam and foi the mean caloiy is 17'J!) incalones JK i uiul of mass This is in good agicement wilh Ihe value ol A ,, inlciied hoin Swaim's meabinemcnt of K v 'J'he mean ol Ihe two is \7'27 Taking Swann't, and Joly's icsiills lonclhu, il niav 1 eoneludi d that the mcabiiicd ^aluc ol K loi an is .iboiil 17'J7 ealoiv pel unit of mass, 01 i 98 ealoi ics pei giamme-molc-euk , al <)' (' and one atmosphcic The same ligme may be laken ,is ncail\ I UK ol ollui diatomic pei manent gases (oxygen, inUogcn, e.nbome o\nh) Theie is conclusu ee\idcnce lhal Ihe speeilie lu-nl ol ll sc n.| S( s rises with the Icmpeiatine 'J'his poml, which is impoilaiil in relation to gas-engines, will be eonsuleied in Ihe in \| aihele S%vann also applied the melhod of eh clue healing lo (Uleuninc' K Jt foi caiboinc acid, and lonnd it lo be 02017 CM!OI\ pei <. Jm ume at 20 C and 02211 at 100 C.i} Jl we assume UK uleol <-haiin ( - with tcmpeiatme to be timfoim liom 0" lo 100", llu liguie lot A at would beO ]9C8, equn nlcnl lo 8 ()(i calories pei H ianiiii( -nidh - cule, in being <U The eoiiespondmg viilnc ol A' ; , is (, (>.S, mid y is barely 1-8 These results aie m I'air agieem. nl willi Ihose oblaiiu-d by Joly in direct measuiemenls of UK spec-die lual ol cnibtniic acid at constant volume. * SAwann,P/^ Tiam A, vol 210, p 231 Aim, U HJL ,,,, L of llm |{, ,i lrt |i AHHI Oommittce on Gaseous Explosions B A Hop lj>i)8. t loc.cil t Joly, Phd Tians A, vol 18J, 1801, p. 73. Reduced to mean calonoa <rc] INTERNAL-COMBUSTION ENGINES 243 164 Variation of Specific Heat with Temperature It was pointed out in Ait. 21 Lhab a, gas might he perfect m bhe sense >f confoinung Lo Boyle's Law and to Joule's Law, so thai Lhe eqiui- .ion PV = IfT is stnclly applicable, and sLill have its specific heat /ary with Ihc tcmpeiatuvc, though Ihcie would be no vanaUon ,vith the piessurc Aii3 r \ analion of specific heat wilh pressmc is due Lo inipcii'ecLion )f the gas In the permanent gases, Lheie is but little departuie Voni the equation PV= JIT except at picssincs niueh higher than hose thai me found in gas-engines Hence (hen specific heat is icaily independent of the piessiuc Even Lhc mixture piodueed by i gas-engine explosion, compusing some water-\apom and cai- )omc acid along with niueh mliogen, conforms to the equation y V RT neaily enough to allow Lhat equation to be applied m alculatmg the tempciatiuc Iioni Hie absence! piessme VlLhough he specific heat ol'such a nuvluic is undoubtedly somewhat gicatei I high piessiucs than at lo\\ pussines, the dilieience is not so onsideiable as to be taken mlo account in gas-uigme calculations On the othci hand, the specific heal of such a mivhiu, and ol lost gases, \aiicslaigcly \\iLh I lie tempeialuie. becoming gicalei s I he tempeiatuie uses, and the citect of llus on the \\oiking of as-engmes is fundamentally nnpoitant In nionalonuc t>ascs, such as aigon or iRlnim, iheie is hllle, if iiy, meicase of specific lical with use oi tempeiatuie, m diatomic ases such as o\\gen 01 mtiogen the mcieasc is consideiable, in asis of moie compk\ conslitiilion, sueh as the Uiatonne g.iscs [_,() and CO,, il is laigei still The pie'bcncc e>l these constituents i a gas-wig UK m:\tuie makes its laic of change of specific heat ith Umpei'ihiK gicatci than that of an The specific heal ol a ris-engme mivtmc at 20(H)C is about 1 S limes what it is at ()C An olnious result of the mcieasc of specific heal with tcmpeia- ue is that A\hen a elelimle quantity ol heat is given to a gas 01 a nxlme of gases as ioi instance by the explosion at constant L)lume in a gas-cngjmc, the use of lempcratme is less Ihan it would i: weie the specific heat lo keep constant, for as the gas geU hotter ich degree of use absorbs more and inoic of the available heat. r r hen the cxpeiimcnt is made of exploding a chaigc in the cyhnelci 'an engine or in any closed vessel, it is found that the tempcratuic itually i cached is fai short of that calculated on the basis of con- ant specific heat, af tci making full allowance for loss of heat to the alls of the vessel. When this fact was first observed it was put 244 THERMODYNAMICS [en down to impeifect 01 lathci delayed combusbioii of the chaise, the suggestion was that a laige part of the heat of combustion was developed gradually, in a comparatively slow process called " after - burning," \\hic\\ was supposed to confciniic aflcr Lhe explosion had spicad through the whole vessel and aflci the tcmperatuic and piessure had iisen suddenly to llie obsci vcd maximum The notion that there is any consideiable effect due to " after-burning" is now abandoned, and it is ricogm/cd that I he facts aic sufficiently ex- plained by icference to the incicasc of specific heat with use in tempeiatuie Measuiemcnts of specific heat, showing this mcicasc, have been made in vanous ways ' by dueet heating, up to high tempciatmes, under constant pressure, b> observing the use of tcmpciatmc in explosions, and also by a method due lo Clerk), in which the gas in an engine cylinder is successively expanded and compicssed several times while the valves aie kept closed In that piocess, the woik done by 01 upon the gas between any two points of the stroke is determined by measuring the aiea under I he indicatoi cuive, and is used as a basis loi reckoning the change of mtcinal energy, \\hile the change of tempeiatine is infciied fiom the cliange in Ihc pioduct of piessuie and \olunie The mUliod can be applied eithei to impiisoned an 01 to an exploded change' It is subject lo some uncei taint} in the estimate that has to be made ol the heat which is given to, 01 taken fiom, the gas by Lhe cylinder walls The results of these \aiious methods ol experiment aie not veiy accoidant Jn geneial the figme got lor the specific heat of a hot gas, when the heating is done by internal combustion 01 by com- piession, is gieatei than when the gas is heated unclei constant piessmc. All the methods arc liable to ciiois which arc difficult to allow foi They agicc m showing that Lheie is an impoitant rise in specific heat with tempeialmc, gieater m Uialomic gases such as watei-vapour or caibome acid llian m mtiogcn 01 an The late of mcicaseis probably not uniform in dialomie gases, it is certainly not umioini in tiiatonue 01 moic complex gases The results aie olten expressed by means of a formula implying a uniform late of increase, K v = (K V ) Q + at, * Particulars ol these, and a vulimblo diHCuasion of Lhe toaultg, will bo found in the Eepoi ts of the Butish Association CoiumiLLoo 011 CJasoous Explosions, fiom 19US See also Sir D Cleik's book on Tltc Gas, Paid and Oil Emjutc, vol i \ D CIcikjPjoc Hoy Soc A, vol 77 vi] INTERNAL-COMBUSTION ENGINES 245 Avhcie (A r ) is the specific heal at C , K v is the specific heat at any tempciatuie /, and is a coefiicicnb that is constant for the paiticular gas Similaily, K 1} = (K P ) Q + at. The coefficient a is I he same foi any one gas as in the formula for K v < since (assuming the law PV= RT to hold) the difference be- tween 7v ; , and K^ is a constant, independent of the tempeiatme Moic piobably, howcvci, I he uitc of mcicasc of specific heat with tempciatuie is not constant, and foimulas of the t} r pc K v = (AT,,),, + at + pl\ K 3> = (A%) () + at + Pi 2 aie leqmied to cxpiess the i elation, especially in gases where there is much change In the absence of nioic exact data a hncai fonnula may scive Loughly, Ihiough a modeialr langc, foi an, nitiogcn, oxygen, and caibonic o\ide, namely A' t = I OS | 001 /, 111 caloncs pci giainmc-molceule .For caibouic acid Langcn ' gnes, as applying fiom 1100 lo 1700" C , a foiniulci (based on explosion cxpciimcnLs) |iu\ aleul lo K = (j 7 H- OO.V2J, ind foi waLei-vapom he gn cs A',, = 1 <) 1 -1- OOl'V, both m c.iloiics jici giainme-molec iilc Tlicic c'.in be lilllc doiibl, ho\\(\i'i, Uial a hncai I'oimula is not i e<i II y applicable lo I lu M' g.iscs I Inonnli ,in\ r \\ idc- i .ingt A leim in - is ictpiiied <is \\ell .is a Imn in /, cspeoulh al Icinpci.ihucs such is aic icaclied in y.is-engiiu's Results colic cltd b\ r I he Hnlish Associahon Commiltcc, for a Lypic.il gas-enoine nuvluic, \\ill be discussed in Ail 108, and li will jc shown thai I hey unolvc a loimula of Ihe Ij-pc K = (A'J,, + / -I- fit- 165 Internal Energy of a Gas What we aic piachcally con- icincd AVI Hi in the gas-engine is not so much the specific heat as a * Langim Xcilt dca Vex nics d&ntacJiei Inyancine, vol 17, 1003, p d22 | Piobalily lias IH too low In an ideal tuatomic gas A',, would ho 3K or 5 905 Ai I 17 !) in water- vapoui it, should bo higher Values of Uio spcoifio heal ol watoi- found by Pioi in oxploRion ovpornnonLs (quoted m Raokiu's "Thonno- and Thermodynamics," Tin Gibson, p 72) irmko A',, noaily G 1 for t = Q 246 THERMODYNAMICS [or. quantity closely iclatccl to it, namely the internal energy W When the charge is exploded at constant volume ils internal encr^v mcieases by the amount ofhcal developed, less what is lost In the cylinder walls. In adiabatic compression the gas gams internal ewii>v t <|m\ alent to the woik spent in compressing it, in adinbalie expansion i! loses internal energ} r equivalent to the work done Hehveen ,mv h\o points a and b on the cm ve of expansion 01 eomj)iessioM in .in .ielii.il engine, if theie is no combustion between ft and b, the equation holds 7T _ TT _i TI/ i n &a Jj J> + '"nb ~l <*:&' wheie E a is the internal energy at a, E b is tin 1 mlein.il c nei<>v -il />, W ab is the work done bv the gas in chan<>m<> fioin slnle a lo sl;de /;, and Q ab is the heat lost to the w.ills dninii> (hat ch.ingc of sl.ih If theie is any inteinal combustion between a and /;, LJCIK i.il ni'; heat repiesentecl by Q' ol , the equation takes ihe nion ^ciu-i.il f01 ' m * Hence if we know the values ot E loi all stales v\e m,i\, l>\ ing the indicatoi diagiam and so nic.isin nn }}',, I'oi .m\ slip, deteimme completely the liansl'ei of lie.il belween o ils ,iiid UK l.d A knowledge of the values of E for the woilunn <;,is I |IM>IIM|I<MI| a lange of temperatuie fiom sav 100 C 1 lo iiooo" (' | S MK M (,, of great piactical impoitance What is w.mled is ieiii\( S|I.)\M M ,, the relation of E to the tempciatuie f The relation between the mlcinal eneig\ K .md Ihe S | t die h, .,1 is that ,,-, r , ,, (IE = A,,r/7 Hence, at any tempcicitme, the slope of (he em\e ,.| l<] .,,,<! y , dE nam ely ^ measures K v , and E - JK v tlT If K v weie constant the enr\e of mteinal <nei..y would !., .1 stiaight line and we should have ,, Here t is the temperatmc on Ihe eenhgmde se.de, ,.,! || u - ,.,. stant of mtegiatmn is /cio if the us.ul eonvenhon |, ( . m | n|ll( ,[ , vi] INTERNAL-COMBUSTION ENGINES 217 icckoning the cneigy of the gas from an arbitraiy staiting-pomt ab C. This of comsc docs not mean that a gas at C. has no in- ternal energy, but only that the staled value at any tcmpeiature is the excess above the value at (compaic Art. 60). If K v = (K V ) Q + at, Oi, if K v = (K ) -I- at + pi*, n lr , . at" # 3 E = (K v ) t+ 2 +'- 3 We may accoidmglv construct a cuive of E and t when nn cx- picssion foi K v is given, or conveiscly find an cxpie^sion for K v fiom a >n en cui vc of K tnul / Fuilhci, when the cinvc of E and / foi a gas 01 mixtmc of oases is diawn, the value of K v at any icmpcratnic is icadily found bv mcasuimg tlie slope of Ihc cin\e Iheic. Fioni llial K niav be deduced by adding Hie g.is-eonslanl R to A',,, namely 1 OS 5 if A',, is cxpicsscd in giamme-caloiies pei giammc-mokcult. (\it 1(>I) In llus wav Ihe latio y of K n lo K v m,iv be dcU'iinuud loi any lempciatuie 1 66 Adiabatic Expansion of a Gas with Vai table Specific Heat In any o.i^ whose specific heat mci rases AMlh llu 1 ttnipcia- tuie, y is nol constanl but bcconus Ii ss ;is Ihe lc inpe i.iline us' s This is t in ob\ ions consc(inence of Ihe f.icl lh.il \\lnlc Ihe specific lic<ils mcicase \\ilh using Icmpeiahne llu diffeituee bt I \\een Ibeiu keeps eoiislanl, since Ihe nas shll conloiins lo Ihe equation l*V ~ RT Ilcnee in alleinpluin lodiawnn adiabalie eui\c lot Ihe expansion (01 conipiession) of such d gas, lluough any gn en laho of volinui's, by means of Ihe ee[iiahon 1*V J consl.ml, \\c have lo do wilh a eonlinuous vanalion in Ihe index; y. When we know the icla1ionof7 lothc lenipeialine we may obtain an appioxunnlion to Ihe euive bv subsliLuling foi the vanable index y a conslanl index >i, chosen so lhal the gas does nol on the whole gain 01 lose ho.iL in Ihe process Foi llus purpose an aACiage value ol y is guessed al and piovisionally used as Ihe value of n. The J'oimula PI'" = conslanL is llien applied lo com pule Ihe work done, and also Ihe iiiuil Lcmpeialure The nulial lemperaLuie is assumed lo be known, as well as Ihe initial pic^suie and volume. It is then seen whether the amount of work so computed agiecs 248 THERMODYNAMICS [cir with the difference between the values of E for the initial and final temperatures If they agree, the gas has on the whole neithei gained 1101 lost heat If there is any discrepancy, ib is to he collected by using a somewhat diffeient value ol n This process of tnal and erroi gives a PV curve winch does not exactly coincide with the true adiabatic cuive but icpiesenls it fairlv well It falls lathei too fast at fast, foi the value of y is less than n in the eaily stages, later it crosses the true cuive and finally lies a httlc above it At the beginning there is some slight loss of heat in the assumed expan- sion with constant n, towaids the end theie is a gain of heat which balances that loss The final tempciatuie in the assumed expansion is a little highei than in tiue achabatic expansion, and conse- quently the woik aiea is a little less 1 -. The process can be made lo gn e as close an appioximation to the tiue adiabatic cuive as may be desned, if we divide the \\holc expansion 01 compiession into seveial steps, and deal in this niannei "uith each step in succession, finding an appiopnate ^alue ol the index // foi each step When a foimula connecting the specific heat with the tempeia- tuie is established, a icLition between F and T dining adiabahe expansion can be obtained as (ollo\\s in leims ol the coefficients used in the foimula, and fiom that i elation the foi m of the adiabal ic cuive can be detei mined Let K L , = a -I- bT -|- cT~ and K ]t =-- a' + bT + cT-, T being as usual the absolute tempciatuie. Since in adiabatic expansion the woik done by the gas is equal to the loss of internal cneigy PdV =-dE=- K u dT . . _ (IT Pdl" Dividing by T, A, T + - ^-- = 0, T , dT RdV n or A T + -y- = 0, 72/F being equal to PIT. But K v = a + IT + cT 2 and K = a' - a Hence a d ?- + bdT + cTrlT + (a 1 - a] ~ = * This piocess of appioxunating to the adiabatic cmvo is doscribor] by Ilopkin^on, Proc In-st Mcch Eng , Apnl 1908, p 443 vi] INTERNAL-COMBUSTION ENGINES 249 Integrating, rT z a loo t T + IT + 4- + (' - ) lop e F = const 2i rr, - hT \ 01 T"F"-e - -const, e being 2 71 S3, the base of the Napicnan logai ithms This equation connects T \vilh V in I lie acliabatic expansion 01 compression of Lhe gas The most convenient way Lo apply it is to \\oik out the value of Ihe cons I ant foi the initial volume and Icmpcialurc of Lhc gas then take another value of the tcmpeiatuic (]OAVCI foi expansion, highei for compicssion) and find V foi that, and so on. Bj diawmg a cuivc of V in iclalion to T it it. easy to find the tcmpcialmc foi any assigned ralio of expansion 01 compicssion, and then to mfei P PV by means of Ihe i elation -^-~ constant 2' Since, by that iclalion 7' = PF a y constant, the above equation for adiabatic expansion 01 compiession may be put in this fonn, P"V" e' ~ ~= const ")i, chminalmg V, P-"'T"e ' --const, which duecllv connects P with T But neithei of these altcinalive 01 ins is so convenient as the fust foi di awing the adiabatic cui\c. 167 Ideal Efficiency as affected by the Variation of the specific Heat with Temperature \Vc saw (Ail 1 r > r >) that he standaid of efficiency known as the "an standaid" assumed onslanl specific heal When Ihe woikmo substance is .1 t>as, 01 Mixhne ol gasis, m \\hich Ihe specific heal me leases \\illi the em])eialme, I lie ideal enicicne> , \\hich the enomc would icach jf Ihe om])iession and expansion weic adiabalic and the explosion weic ompleled al Ihe dead-point without giving up any heat lo Hie ylmdc r walls, is neee'ssanly much less m consc()iience of the lowei Maximum e>l lempeiatuie The ideal efficiency with vanablc pecific heat ma}' be elclei mined Avhcn we can cxpicss /v,, as a nnclion oi Ihe Icmpciatmc, 01 when we know Ihe iclalion of the uLcrnal cncigy to Ihe lempcratmc, but it elocs ne>t admit of any implc cxjjrcssion It may be computed by woikmg out Ihe lelcal idicaloi ehagiam, with adiabalic curves, assuming a heat ofce)in- 'iisliem appioj)iialc lo the given mixtiuc, and then comparing he woik which Ihe diagiam rcpicscnts with the assumed heat of ombusLion 250 THERMODYNAMICS [en When we can expiess the specific heat as a function of the tem- perature we can calculate the ideal efficiency a^ follows Taking as befoie K p = a' + bT + cT\ assume an initial tempera tin c T at the beginning of compiession. This, which is called the suction temperatme, is gcneially taken as 100 C. Then apply the adiabatic equation to find T l , the tempeiatme at the end of compiession Next calcu- late T 2 , the tempeiature aftei explosion, by equating the change of E to the heat given out by the binning of the chaige Then again apply the adiabatic equation to find T 3 , the tcmpeiatuic after expansion. When these four tempeiatuies aic known the coric- spondmg values of E aie determinate. The woik spent m compies- sion is E-L E Q , the work done in expansion is E^ E^, the heat supplied is E 2 E! Hence the ideal efficiency is E- E, - (E, - E Q ) E 2 - E, When this calculation ib made foi a typical gas-engine mixture the lesult is to give an ideal efficiency about 20 pei cent, less lhan the air standard, foi such latios of compitssion as arc in common use ' . We saw m Ait. 155 that with constant specific heat and thcie- foie constant y the an-standaid efficiency was A good appioximation to the efficiency of the ideal cycle with vanable specific heat is given by the cmpincal foi mi i Id Heie the index 3 ma}' be said to icpiesent a gencial average of Lhc values of y 1 throughout the action (assumed to be adia- batic) of a typical gas-engine mixtuic. (Compaic with Art. 168 \\heie values of y aie gncn for vanous tempeiatuies.) * See Mr H E Wimpens' tex:t-boolc ol Tha Inlet mil Compilation Engine, Chap iv, wheie results aie given in detail, for various values of ? The specific heats aie theie expressed as linear functions of the tempeiature (the term oT & being omitted) For reasons \\hirh \vill appeal in Ait 168 the term m T- should be retained vi] INTERNAL-COMBUSTION ENGINES 25] It should be noted however that the ideal efficiency, for a mixture in which the specific hcaL vanes, depends not only on the latio of compression but also on the strength of the chaigc. A stiong charge gives on combustion a highei temperature than docs a weak charge, because of the greater development of heat, and it also pioduccs a mixture which contains a laigei proportion of waler- vapoui and caibomc aeid Foi both of these icasons Ihe influence of va nation in specific heat, in i educing the ideal ciricicncv below that of the air standard, is gi cater foi a sliong charge than for a weak one In an example given bv Ilopkinson *, where the ratio of compression was 6 37 and the an -standard efficiency was Lhcielbie 0522, the ideal effieicncv (allowing for Aanation of spceilic heat) was computed to be 121 foi a mixture containing S 8 pci ccnl. of coal-gas before combustion, and 0301 foi a stiongcr mixture containing 11 '! pci cent. Fiom these figuies it would appeal IhaL a mixtme eontaining about 9', pci cent of coal-gas would ha\c an ideal efficiency 20 |)ci cenl less than the an sl.mdaid To dctcinnne the ideal c(hcicnc\ is a maltci of impoitancc because, by compaimg it \\ilh I IK clheicncv aclualh ie,ih/ctl, we aic able lo sa\ what is the niaigm of impio\emcnl foi leducmg the Iheimod) iianue losses that ocein in the aclion ol a icaJ engine 168 Curve of Inteinal Energy for a Typical Gas-Engme Mixture The Untish Yssocialion Committee on Gascons Explo- sions QI\C in I hen fnsl Ucpoit (100S) ,i cui\ e of inleinal cncigvand Lcmpcialuic loi <i l\pical gai-cnginc mixlnie, namely the nu\tuic used b^ C'kikin Ihe cvpciimcnls to \\hich u I'eicnce has been made This mixluie was the pioducl of combustion of a cliaige ol one part by volume of coal-gas to about nine paits ol an, toclhci with Ihc bmnl gases in ihc cleaiance space it contained 5 pci ccnl by "volume of caibomc acid and 12 pci cent ol walei-\apom, Hie icmaming S3 pei cent being made up of mliogen and suiplus oxygen The cm vc, which is ic])iodueccl in fig 82, is based pailly on Clcik's expeiimcnls, m which the hoi mixtuic was expanded and compies- sed in an engine, paitly on earlv cxjilosion experiments by Mallard and Le Chatehei [ and Jatci ones by L'ingcn |, and paitly on duccL mcasuicmeuls of the specific heat of gases at constant pressme, * Proi Inil Mcih Kng , Apiil, 1008, p 425 ) Mallaid and Lo Cliatolioi, Ami dca Mines, Ib83, p 274 f Langon, loc cil, , also Mittoil ilbcr Foisc/iiingsarbcitrii, font Vt> dcttlscfi Ing, Lleft 8, 1903 252 THERMODYNAMICS en. o o o 9fnodioiu-9uiuivj$ jdd o o o CN O O O o vi] INTERNAL-COMBUSTION ENGINES 253 by Holborn and ITenmno* The curve shows the value, in relation to the tcmpeiatuu, of E cxpiessecl in giamme-ciiloiies pei gramme- molecule that is to say in o rammc-ealories foi A volume of the mi\> tuie equal (o 22,100 cubic ccnLimctics at C and one atmosphere To reduce K to fooL-pounds per cubic foot multiply by 3 00 | The curve as on^inally given in Lhc Committee's Report stalls from 100 C , which is laken as the /eio in icckonmg E As here leproduced iL is extended down to C , and E is icckoncd from thai poml in accoidancc with Ihc convention alicady mentioned A careful examination of the cuive shows llut no foininla of the type K = (A",,),, -I- / will lit it K v must mciease slowly at low tcmpeiahues and fas lei at high tempeuiturcs to gi\c values of E thai will ai>iec with the cm\e It is, howevei, well rcpiesented by the loimula K = ,- 2/ + f) 00() j fji j + 000()0()2/ J , which couesponds (Art 105) lo K u = 5 2 + OOOSW + OOOOOOO/ 2 Values of E measincd liom Ihe cuive and calculated horn the above loimula aie compaicd in the table below Intt'tnal Kn(i<iy <>J Gat*- Engine A J'J in caluncs poi litiiiniuo molcuilo Mioa^iiind hom IV nip ' tin ( iu\o 200 1000 J0 r ><) K)0 21()0 2l()2 1)00 ,{.{20 .J.JI8 800 1, 1510 4/5 1000 r >SIO f>8.JO 1200 7200 7205 I JOO 8(>,SO S(>72 11)00 10,210 J 0,2 10 ISOO l!,')20 ll,')20 2000 13,720 13,720 As was pointed oul m Ail K5, the slope of Ihe cmve mcasuies K v at any poml The initial slope, foi t = C , is 5 2 calones pei clcgicc The slope foi t = 2000 C is 9 32. Within I hat range the >peeilic heat has increased by nearly 80 per cent. At 2000 the gas * JLolborn and Honmng, Ann (hi Phi/s , vol 23, 1907, p 809 | In the Report the volume of the gramme-molecule is given as 22,250 o cms ind this factor ns 3 96 Thoie appears to bo some enoi m both of those figures 254 THERMODYNAMICS [en. contains 30 per cent, moic inteinal energy than it would contain if I he specific heat were constant at 5 2 The specific heat at C agrees faiily well with the value which we might calculate from the known composition of the mixture, using the figures given m Ait 1 6 V namely C 7 foi CO_>, 5 9 for H,O, and -1 9S for the remainder of the mixture Since Ihe proportions by volume wcie 5 pei cent of CO,, 12 per cent of II 2 O and 83 per cent, of other gases, we should expect K u to be 5x67 + 12x50 + 83x198 _ foo = o 18 If 6 1 weie taken foi II 2 instead of 5 9 the agreement would be exact (See footnote on p 2 l-l ) Values of K v , K }) (taken as equal to K h 1 985), and of y, lor this mixtuic at various temper atmes are given below Temp U C '' 500 100U 15UU JOOU' K a 5200 5780 OObO 7840 9 320 K,, 7 185 7705 S (5-15 9825 1 J 300 y 1 38 135 1 30 1 25 1 21 The importance ol the t- tenn in Ihe foimula foi K is obvious, especially as alfcctmg the values at high lempci atmes The uses to which a cm\e connecting the inteinal cncigv wilh the temper aim e can be put in the analysis of an engine lnal aie illustrated m the next aiticlc 169 Action in a Real Engine. Analysis of the Indicator Diagram The indicator diagram of an internal-combustion uigmi sei\es not only to mcasuic Ihe work clone, but to liace I he changes of temperature thioughout the cycle, piwided Ihe lempuatine at one point is independently ascci tamed, and piovidcd also Iheie is no leakage of the woiking substance Undei noimal conditions of good running., the leakage is negligible The gas itself serves as a theimometci, since T vanes as PV Thus if we compaic any IvVo points a and & between which there is no chemical change, /-> 77 p T r * a f <i _ *_i f _p T ~ T ' * a L l T P V from which 2' 6 =-"-. - 6 * a' If however combustion has taken place between a and ft, we have to allow for the chemical contraction by introducing a factor a V,] INTERNAL-COMKUSTION ENGINES which cxpicsscs the ratio of the specific volume after combustion lo the specific volume befoie combuslinn This makes T p r 7' = -' ?J _ '' aJt ,J\i Thus if we know Ihe "suction tempeiatme" T of the charge when compiession begins, it is easy to imd fiom I he indicator diagiam the lempciaturcs at any oilier points, such as the end of com- piession, the end oi' expansion, and the point of maximum piessure These Icmpciatuic's are of couise in all cases the mean temperature throughout the whole volume ol the gas Take foi example fig S3, an mdicatoi diagiam m one of Ilopkmsoifs tesls ' ol a gas- L'ngine \\ith a coni])iession latio of (j 37 In this instance the L>as-nuxtuic al Ihe beginning ol -ompiLssion had a lempeiatiue jf 100 C , and a piessiue ol L 1 7 pounels pel sq inch Itcon- amed 11 pci eenl ol coal-gas mel the chemical conliaclioii \as 3 pei cent , making cr -- ( )7 \\hcn Ihe ehaigc was igmleel he combuslioii \\as so lapid lhal Ihe piessiue lose, lo ils maxi- num beloie the |)iston had ino\ed pciccplil)l\ loiwaid L r siug Ihe ame sulhxcs as m lig 79, \\e see horn Ijic diagiam lhal P l al he end ol compiession was 105 pounds, /^ aflei explosion was S2 pounds, and J\ at the end of expansion was 10 > pounds, as iieasuicd by piolongmg Ihe expansion cuivc to tiie end of the tioke, beyonel Ihe point al \\hich the release-valve began to open lence (using absolute tempLralmes), 117 x (> 37 500- 400- J 300 1 200- 100- J'V r s J lndiuaLoi Du^um timiia(Jn> l')()s(ii l) ,,L ll ,,u,,) ]> 1' 1 ()' T rit * 2 *- '1 ~r al\ 97 x 105 '*- 1(J8 x '^ 5 x G J7 - _ ^_ _ * P/oo Jnst JUeUt Ena 1008 ID 420 = 1080 C 956 THERMODYNAMICS [en Or, calculating ducct horn the initial stale, T = 3 = 1,95 J ff P 97 x 1 I 7 Hence, if a cuive of 7? and T Ibi the mixtuie is available, it is easy to find the values of E at successive stages, and, by compamig them with the work done fioni stage to stage, to detci mine the heal oiven to 01 taken 1'iom the walls dining compiession, explosion, O and expansion In the example quoted, Hopkmson lound that the indicated woik lepiesented 33 pci cent of the heat ol combustion, and thai the gases at the tenipeiatmc oi icleasc (7' 3 ) cauicd oil ,'39 pci cent , leaving 28 pei cent as the net loss bv lachalion and conduction Most of this Jast item is taken up by the cnculaling walci of the watei -jacket which is used to keep the cyhndei cool enough loi lubucation. The 33 pei cent convcited into woik icpusenls an efficiency-ratio of S3, lor the ideal efficiency nndci tlu sc conditions ot compression and mixtuie stiength was 10 With a weakei mixtmc, containing 8 5 pei cent of coal-gas, he found that a laigei fi del ion, namely 37 pci ecul , ol Lhe luat ol combustion Mas conveilcd mlo indic.itcd woik This concsponds to a InghcL clucicncy-ialio, namely 0<S7 As llopl mson obseixcs "the wcakci niixtines, in tidchlion to gi\mg a Inghci ideal c HICK no , come neaiei in pi ac lice io leahsing that ideal " This is bce.msc they lose iclalivcly less heat to Lhc walls dining t \plosion ,ind c \- pansion ''The diifcicncc is siifhcient to eounlcibalance <m mllu- ence lending the othci way, namely Lhc moic uipid combustion of the slionger mix lines " The less ia[)id combustion ol weak mixtures is appaient milieu mchcaloi chagiams it shows ilscll b\ the maximum piessuicoceuiung lalci and with amoie lonnded lop In extreme cases the combustion is piolonged Unoughoiil (he \\hole expansion stiokc, and I lie exhausl gases conlain unbmnl piodncls Incidentally these figures illuslialc the most obvious weakness of the oidinaiyinteinal-eombustion cycle that the latio of expansion is no gieatei than Lhc uilio oi compicssion Consccmcnlly Lhe pio- ducts of combustion aie chsehaigcd at a high Icmpeialme wilh much unutilized internal energy. Attempts have been made to avoid this loss, notably by Atkinson, who designed Joims of engine in which the expansion stioke was much longei than the compulsion stioke The advantage, m icspcct of thermal elficicncy, was con- siderable, but the complication of Atkinson's engines stood in the i] INTERNAL-COMBUSTION ENGINES 257 'ay of their success, and in modern engines the theimodynamic d vantage of high expansion is sacrificed to mechanical simplicity. 170. Measurement of Suction Temperature. In the above nalysis of the diagram it is obvious that the icsults depend on ic accuiacy with which the temperature is known after the admis- on valves die closed A icsistance theimomctci of ^ eiy fine wnc ill serve to measure it when furnished \\ith an electncal contacL i the shaft by means of which its indication is given only at the ghl moment in the stroke. But such a thermometer will not stand ic high temperature which is i cached in the explosion Messis illendar and Dalby 1 have devised an ingenious plan for ovci- >nung this difficulty In their device a fine wire of platinum foims LC theimomctci. It is fixed to a tube which slips llnongh a hole in ic spindle of the admission val\ e, and piojccts into the gas The be has a valve-shaped end which closes when it is drawn back, e wne is then scieened fiom tlic action of the gas, and no gas can cape. Bcfoic the end of the compulsion sliokc it is diawn back y means of a cam on tlic "salve-shall), and lemanis scieened nmg explosion and expansion. It is again piojcclcd into the gas foic the compicssion stroke begins The \al\c-slufl of lliccngine, nchiolales oncefoi each cycle of foui sliokcs, cauicsacam whicli mpleles the clectnc ciiciut of Ihc thcimomelei al the j)iopci ml, just when the admission ol the chaige is complete and com 3ssion is about to begin 171 The Process of Explosion Much light has been tlnown whal takes plate when a gas-engine nuxtnie explodes, by cxpen- i nls m which gas mixtmes have been exploded in closed \cssels constant volume, with devices lor icgislcimg the use of picssuie iclation lo (he time, anel also the piogiessive changes of tcmpcia- e al vanous points within the vessel The experiments of IIop- ison on explosions of coal-gas and an should be specially icfeiied in this connection |. _,ct an explosn r e mixture, homogeneous and at rest to begin with, ignited at any point A flame spreads in all directions from the nt of ignition, Uavellmg at a late which depends on the pres- e, so that each portion of the mixture ignites in turn, the most Proc Roy Soc A, vol. 80, 1907 See also the Seventh Report of the British jciation Committee on Gaseous Explosion (1914) B Hopkinson, Proc Boy. Soc A, vol 77, p 387, 1906, also vol 84. ra ICfi 258 THERMODYNAMICS [cir. distant portions last When the initial piessmc is bhal of Ihe atmo- sphere, the flame may tiavel at the rate of only about five feet per second to beg in with, even in a nch inixtme such as one of gas to nine of air. The late depends on the richness of the imxtuic as well as on the piessuic in. a weak mixtme it takes much longci foi the ignition to spiead thiough the whole volume. When the initial picssure of the mixtme is high the ignition flame travels much fastci. The poition which is fiist ignited, close to the ignition plug, bums at neaily constant piessure, being sin rounded by a Luge clastic cushion of uniginted gas. Its combustion is practically completed befoie the piessme has nsen Then Ihe spicad of (he (Lime bungs more of the gas into action, the picssuie uses, and the portion which was hist burnt is compiesscd This compression is nearly adiabatic its effect is to laise Lhe temperatuie of that poi I ion much aboi* e the temperatuie to which it \\as brought bv combustion, and above the tempeia tine whichis reached in combustion b) the outlying paits of the gas, which aie compiesscd beloic thc\ become ignited. In Hopkuison's experiments a mixtuic. of nine pails ol gas lo one of air was, filed at atmosphenc picssuie in a c\ luuhical N essd with a capacity of about 6 cubic feel It was ignited b> an elecliic spaik at the centre, and dc\ eloped a maximum pussmc ol about SO pounds pei sq inch, uhich. was i cached a quailei ol a second al'lei filing The tcmpciatuic was obseived near the ccnlie- antl al olhei points On ignition Ihe tcmpciatuic at the ccnlu lose vei y lapidly to 1225 C , while the picssiuc icmamcd neaily constant In the latci stages of the explosion, \\hen the biunt gas al UK ernlie \\as being admbalically compiesscd, its tcmpeiatiue lose abo\c the melting point of platinum, piobably to 1000 This is a hi"he i tcm- pciatuie than was i cached in the outlying portions, \\luch xu-u- hist compiesscd adiabalically and then heated by combustion \\ hen the maximum picssuie uns i cached, the mean tempeiatuie infeiied from it was 1600. Hopkinson concluded that even m a vessel impei vious to heat, the poition of the mixhnc fhst hied would be hotter than the outlying portions by about 500, when the combus- tion of the whole was practically complete An interesting conse- quence, pointed out by him, follows from the fact that when the combustion is complete the gas is not in theirnal equilibrium Imagine the loss of heat to the Avails to be aucstcd while the burnt gas settles into equihbmun of tempeiatuie. If the specific heat were constant this settlement would make no difference to the mean tempeiatuie and therefore no diffeience to the picssuie But 'ij INTERNAL-COMBUSTION ENGINES 259 )ecause the specific heat of the hotter poitions is higher than that )f the cooler poi Lions, the tcmpeiatuie which the gas assumes when t is equally hot all through will be somewhat higher than the Dicvious mean, and theic will consequently be some use of pi ess ure is the gas settles into thermal equilibrium These cxpcnments go to show that Ihcie is no substantial imount of "aflcr-bmiung," and lhal the e fleets formerly ascii bed o altc'i-})iniiing are clue lo I he mcicasc of specific heat with tem- >eiatuie Piaclieally Hie full evolution of heal in eacli poition of he gas takes place at once when I he (lame i caches th.it poition, ml there is some dclav in completing the ignition of those poi lions /Inch aic in pioximily to the cold walls, especially when the nxtuie is weak In explosion experiments with weak mi\tmes the spread of the amc is much slower, so slow indeed lhal it is laigcly alfectcd by >m ection cuiients sc t up by I he ignihon of I lie gas ncaiest lo I he )<uk The gas in Hie up[)ci pail of I he vessel may bt completely mted \\hile the Io\\ei pail ol I h( \ ( sst I is still lull ol unbiimt >s By stilling I he conlenls ol Ihc \cssel so lhal the gases aie i motion when the spaik passes, a much moie i.ipid combuslion ' I he whole can be seemed 172 Effect of Turbulence This el ft c I of tmbiiltnce in omolmg i.ipid ignition ol live \\holt conlenls is fell, though lo less degiet, m shong mixlmts ,is \\tll as \\eak mixtmes II is, C'li i k has pointed out, an nn poi I an I f.ietoi in I lit 1 woikmg of an lei n.il-combiislion ( ngine \\'lu u a JK sh chaige is diaun in and mpicssed I he g<ists aie slill m moit 01 kss \iolenl nioliou <il c moment ol ignition Tins h<is I he yieal adxanlaj^e lhal coin- islion is lapitllv piop igalcd I hioughoul Hit chaige and (he ma\i- nm pi ess i ne oecius rally in I he expansion slioke. Ck i k obsei \ etl at when Hie e \plosi\t chaii>t in a gas-engine A\as nol died aftt'i e fust eompiession, bul \vas fired al((i llnee successive compics- 'iis, so lhal (he liubulence set up on its cntiyhad lime m pail s\ibsidc, the process oJ combustion was generally prolonged, th the icsiill ot giving a lint diagiam and a wasteful action In a >h-specd engine the whole expansion stiokc may take only ouc- entielh of u .second, or less, and the explosion is over in a small ction of that time this would be impossible were it not foi the 2ct of turbulence in causing the flame to spiead quickly through cylinder 260 THERMODYNAMICS [en. 173. Radiation in Explosions In closed-vessel experiments, the maximum of pressure is i cached a little before the combustion ib complete, for it occius when the late of loss of heal by ladiation and conduction to the walls just balances the late at Avhich heat is being geneiatcd in the gas Hopkinson has investigated the effect of radiation by comparing the late of cooling in an explosion vessel which was lined with highly-polished sihcr, with the Kite m the same vessel when its inner surface was blackened '. The late of cooling alter explosion was notably gi eater when the walls were blackened, and the maximum press me wo,s less foi chaises of the same composition The rate at which heat was lost Lo the polished walls was on the aveiage about two-thuds of the idle of loss to blackened walls It vaiied with the exact stale of the pohbhcel surface, which Avas nevei perfectly reflecting Ilopkmson conclu- ded that of the heat given by the gas to the walls of a blackened enclosuie dining the hibt quai ter-.seconel aftei maximum picssuie, at least 80 pei cent is radiant heat, and possibly a good deal more, foi the leflectmg quality of the polished walls may ha\e been mi- pan ed by the deposit of a film of moistmc al an eailv si age ol the- cooling Fiulhei experiments, in which Ihc \cssc'l was lillcd \\ith a fluonte wmelow to allo\v the laclialion to (all on an absoibiiig sci cen outside, confinned the view that lachafioii ace-omits foi moic than 30 pei cent of the \\holc heat loss. Its el fee Is \ve-ie slill per- ceptible aftei the tempciatine of the gas had Jallcn to 1200 In latci mcasuieriients by \Y. T. l)a\'id I In 1 lotal loss b\ radiation attei the explosion of a mixtuic ol coal-gas and an in a closcel vessel, was lounel to be about 25 pei cm I oJ Ihc \\hole heat ol combustion The late at which ladianl eiiugv was emitted, thiough a fluonte wmelow, was giealesl a hi lie be-lou- the piessme i cached its maximum it Cell e>l'i lapielly as Ihc exploded gas cooled, bill ladiation could slill be delecle-d when the tempcialuic had fallen below 700 C., about a second allei the chaigc was fhcel | The eneigy of the laduition Jiom an exploded gas-engine mixtuic is due almost wholly to two 01 throe bands oi rays ol definite wave-length, coirespondmg to much slowei vibrations than those which pioducc the visible spec li um. The existence of these bands may be demonstrated by examining the heat * PWL Roy Soc A, vol 84, 1910, p 155 ISee also the Third Repoit of the Biitish Association Committee on Gaseous Explosions, 1010 | W T David, P7iz7 Tiana A, vol 211, p 375 Phil May Fob I'UIiiuulJan 1920 i] INTERNAL-COMBUSTION ENGINES 261 r huh is radiated from a gas Maine when il is made non-himmous V using a 13unscn bmnci Experiments with such J lames show that when hydrogen is buinL j foim waLei-vapour niosl of the ladiant heat LhaL is given off is i a hand with a wave-length ol aboul 2 S/^ 1 , but some lias a longer 'live-length also llial when caibonic oxide is burnt most of the uliant heat is in a band wilh a wuvc-laigth of about 1'i/j,, but >me is in two bauds whose wave-lengths ate about 2 7/j. and ehvcen 1 I- and Ifyi In one ease I he ladialion comes fioin vibrating lolcculos of II_,O, in Ihc olhei fiom vtbiahng molecules of (.'0, ll is also found lhal cold CO 2 absorbs sliongly Ihc radial ion tiom CO llamc, and \\alei-vnpour absoibs slionglv Ihc ladialion horn hydiogen (lame It may be concluded I hat the modes of fiec ibiahon of a molecule of cold CO, or wate-i-v apom ha\e penods Hicspondmg to the chic I \\a\e-Ienulhs which the gas gi\ cs out lien il is so violently agitated as lo become a sou ice of radiation. his happens \\hcn Ihc moleuilis aie foimed l>v Ihc coming )gclhei of then constituent aloms It is liulhei found | lhat a mixed 01 compound gas binning lo im CO_, and If/) gixts oul bolh wavt -lengths (I \JL and 2 S/t), id tluil Hie whole cncigN it ladialcs is Kju.il lo the sum of Hie K igies scpaialclv compiilcd lot I he molccuks of II 2 O and CO^ i.il aie foimid b\ ils combustion Foi e(|iicil volumes of !!_,() id C'O,, at the saint ll.imc Umpciahnc, the ladialion liom CO^ )|u.iis lo be .iboul "2\ liuus I h.il liom IF^O These icsiilts point let I he conclusion lhal \\heu a gas-engine ixluic is In eel, Ihe c'lu igv I h.il is ladiali d conn s almost cnliidv om molt cult s ftf ('()_, ,mel !!_,() in Ihe biiiul g.ises v ei\ hi tie ol it >mes liom Ihe mliogen 01 lhe i suiplus o\\gcn 174 Molecular Energy ot a Gas Aceoiding lo Ihe kinetic icoiy of gases, the mlcin.il cne'i^v E ol a gas is made up of Ihe 'inmumcable cncigies ol its mtilcenlcs, anel t.ich molecule m,i\ , ge'iiei.il, hav e communicable e'neig^y of Uiese- lluee kinds (1) Kneigy ol livmslalion of the molecule as a whole, (2) Eneigy of loUihon of Ihc molecule about an axis llnougli ils eentie ol mass, (,'J) Eneigy of vibialion * fj. staiKla for Jinlljonllis of a inolio TIio wave loiiglhs in tho vmiblo spoclium igo liom about 30 lo 11/j. \ R von lIclmJiolLx seo Hie Thud Koporl of Ihe BuLish Association ComnuUoo 202 THERMODYNAMICS [en It is to energy of Hie first kind that the picssmc of I he gas is due The kinetic theory shov,s ' LhaL, in a gas foi winch PV = RT, the energy of translation is ]RT. The piessmc, in kinetic units, is mmieiicnlly equal to two-thirds of Lhe encigy of translation of the molecules in unit volume of the gas When Iho gas is healed, this encigy increases in dneei propoition le> T Hence \( nil Ihc mteinal eueigy of the gas wcie in this foim we should have E = \RT (leckoning E from the absolute zeio of lunperatnrc), and Ihc specific heat would be constant. K v would I hen be equal lo }]f, which would make K lt = r jR, and y = r \ oj 1 007. This is ncaily tine ol actual monafomic gases in such gases E consists cniirelv, 01 almost entnely, ol cncigy of tinnslaiion of the molecules The second kind, energy of lotation, becomes an important pait of the whole when the molecule compnscs l\\o 01 moic aloms \Vc ma} conceive the molecule of a diatomic gas such as ()_, or N 2 to consist of paucd atoms held at a dcfunlc distance apail like the hea\y ends of a dumb-bell Such a stiucluie may, in the eouise ol its cncounleis, aequne cneigx e)l lotahon about any axis pei- penehculai to the line pining the I wo aloms, bill not about thai line In addition to its tlncc degiecs ol Cieedoni ol lianslalion it consequently has tttoty/fc/ji't 1 dcgices ol I'leeelom ol lolahon, hence fncmallaic effccli\e out of the six elcgiccs ol lieidom \\hich it possesses as a iigid body. According to the kinetic Lhcoiv the encounlcis bel \\een the mole- cules, when the gas is m a stead} state as lo picssiiu and lempeia- tuie, cause the cncigy of lianslalion anel lolalum, (]) and (2) togelhci, to become equally ehvielcel among as many ol lhese v six degiecs ol fieedom as aie elftcliye Hence in a ])eilect eliatonucgas, besides Iheeneigy of Iranslalion, vhich is \R r l\ llieic is .in amount of cncigy ol lolalion equal lo RT due to the two heedoms ol lotation, making ]RT in all foi the fi\e effective dcyices of I'le'edom. Consequenlly m such a gas, il theic wcic no energy except what is compiiscel m (1) and (2), we should find E=]RT, the specific hevt would be e-onslanl, K v would be 47?, A' 3) would be ]R, and y v\oulel be I 01 1 I These \ allies agicc \\ell with those Jounel in actual diatomic gases biich as nitiogen en an, so long as the gases aie ce>ld Uul, as * See Appendix. I f. vi] INTERNAL-COMBUSTION ENGINES 263 we have already scon, Ihc specific heats become distinctly higher at high temperatures and y becomes less This means that, in addition to items (]) and (2), Ihcie is in these oases some energy of vibration (8), the amount of which is insignificant at low Icm- pciatnics, but becomes compaiaiivcly imporlanl when the gas is highly heated. It does not increase proportionally to T but in a moic rapid latio In Inatomie gases such as II 2 or C0 2 , and in gases of a more complex constitution, thcie arc Ihiee effective freedoms of rotation as well as llnec (Vcedoms of translation, making SIK in all, between which the cncigy compiiscd under items (1) and (2) is equally shaied Thus items (1) and (2) account foi an amount of cncigy equal Lo ( \RT If theic wcic no more, name ly no encigy of \ ibia- tion, (he specihc lu il would be constant, K H would be 3A\ K fl would be I A', and y would be ', 01 1 ,'333 In walei -vapour and caibonie acid the \aliu of y c\ en al low Icmpcialmcs is less than 1 333 in walci-\apom il is about 13 and in caiboiik acid il is a little lo\\ei (\it 103) Fiom this, and also fiom the lad thai modcialc healing c'onsidi i.iblx laises the spec'ilic heat, it m<i\ r be m lined that even at low lcni|)i tal mcs I hi moliciilis of these gases h<i\ r c some eneig\ ol \ibiahon [Is pio- poi lion lo I he \\ hole em igy is me i east d by hi a I ing the i>as The amount b\ \\huli the cnugv ol \ibialion augmuils Ihc specific heat in an\ g is ma\ be mleiied hoiu I he \ alue ol y il \\e assume the gas-law P\' = RT Lo appl\ Take loi mslance a tn- alomic gas A,,, il then \\eie no \ibialion, would bi {/', li I nR IK (hi amoiinl bv \\liuh \ibi.ihon.il cn< m\ mcieases il Thin K r - (.3 |- n) If, A,, - (I -I n) It, fiom which y -= ( I | )/(3 -|- //) Suppose that y has I he \alue 1 .30 instead of 1 ,3.33 I Ins makes n J, and the spieiln heal A' 7 , is thc'icfoie 10 pei cent giealcr because 1 ol vibialion The value of n mcieases with the lempcia- luie At the lempeialuic i cached m a gas-engine explosion y for CO 2 is piobably nol much moie than ] 1 1., which would coiiespond lo a spce-itie 1 heal approaching (>A' (See Ail 22I-.) The phrase "c'licigy of vibration" is to be understood as including all the kinds of eneigv which the molecule may acquire, in the course ol its encounters with other molecules, except eneigy of rotation as. a whole and energy of translation as a whole All such forms of energy arc internal to the molecule rtsclf they may be due Lo 264 THERMODYNAMICS [en relative motions of its paits 01 to electucal clistiubanccs within it, 01 within its atoms. It is lo encigy of vibiation I ha L the radiation 9t given out by a heated gas is altnbntcd. When a gas-engine mix tine is fiied the energy generated by the explosion is at first concentrated in the newty-formed molecules of C0 2 and H 2 and spreads to the other molecules as a lesult of enconnteis. We may conjectuic thai Jl is at fust mainly vibrational, and the enconnlcis transform part of it into encigy of translation. It is eleai that the newlv-foimed molecules possess much moie than then noimal piopoilion of energy of vibrdlion, much moie, that is to sav, than Lhev Avould possess if the buint mixtme weie kept wiihoul loss of heat long enough to let equilibrium be attained between the different kinds of encigy, or were le-heated to the same tcmpeiatuie altei being cooled Some time, peihaps only a veiy shoit time, must elapse bcioie a condition of equilibiium is icached. II the gas weic enclosed, aftei combustion, in a vessel impeivions to heat, \\hi\c this process is going on, the eneigy of tiansLilion uonlcl incicase at the expense of the eneigy oi Mbiation, and the tcnipciatme would Ihcicfoie use though the total eneigy undergoes no change So fai as il goes, tins piocess of attaining equihbiiuin lias an c flee I like conhniicd combustion or "aftei-binnmg." The time taken to reach cquili- bnum is not known. If the pioccss is not \eiy soon oonij)lcled jl may account foi the fact that mcasuicmcnls ol specific lie.it made by means of an explosion in a closed ACSSC! gi\c \alues SOUK \\hal gicatei than those that aie got when the gas is healed in oilier ways It has been suggested that the molecules of a hoi gas emit lachation mainly when they unckiyo stiuctiual change If llns view be eoncct vc should expect a gas mixlinc lo rachale moie eneigy immediately aftu explosion lhan when it is maintained at Ihesame tcmpcialine, 01 ic-healed to the same tcmpeiatuie aflei cooling Hopkinson's and David's experiments show lhal in an explosion I he gas continues to rachale for a second 01 so alter maximum piessuie This may only mean that the special vibialions (special m violence 01 in kind) that aic set up dining the a,ct of foimalion, to which ladiation is ascnbed, subside lalhei slowly Thcie is in any case an action going on m all hot gases, Ihat Lends to maintain such vibiations, namely the bicakmg up of some mole- cules by exceptionally violent encoimteis, which is called dissocia- tion, and their subsequent re-foimation. vi] INTERNAL-COMBUSTION ENGINES 265 175 Dissociation In airy gas, however homogeneous, and at any lempeiatuic, the molecules at a given instanb have widely vaiious speeds Some ol' the encounters may be so violent as lo bieak up compound molecules, sepaialing them into paits which after a time meet ficsh pailncis and le-combme The probability of such chsmptive encounters is ob\ lously grcatci the hot lei Lhc gas is In a hot gas in equilibiium, a pioccss of dissociation and ie-combinalion goes on continually, to an cvlenl depending on the tcmpcialme, with the result lhat al in\\ instant a certain piopoilion of I he gas is in (he dissociated stale The piopoition dissociated depends also on Hie picssiue, at high prcs- suie it is less than al low picssure, foi the same Icmpciatmc Accoidmg to measmements by Ncrnst and othcis the amount of II 2 O dissociated, undei a piessure of one atmospheie, is baicl> 2 pci cent at a tcmpciatmc ol 2000 C , baich 1 pei cent at 1800, and 002 per cent, at 1227 C Al a pitssinc ol ten almospheies these numbcis aie about halxcd In C'O 2 at one almospheie, the piopoition dissociated at ](>f>0 C isaboul 1 pei cent and at 1200 about 03 pei ccnl Al such tcmpeialincs Ihcic is })iobablv no sen- sible dissociation in mliogcn These dailies au ope n lo some doubt ' , but it they can be accepted as applxmg to I IK conditions ol a gas- engine nu\line a fit i explosion (conehlions \\lueh <ue not those ol c quih bi mm) il appeals that dissociation pla\ s no conside lable pail in that action So lai as it has <my ellecl it letluecs xeix shghll}, I he chc mica 1 con 1 1 action, by si i bsl i lu ling some molt ciiles ol II, and 0, loi molecules of ILO, and some molecules of CO and ()_, loi mole- cules ol ( O 2 , (01 Ihe same icason il i educes slightly I he immedialc r lcvclopnienl t)l Iheimal eneigv, leaving a sm.ill piopoition of the ivailable chtimc<il encigy of the g.iseous fuel lo be ele\ elopeel latei, is the piopoilion of dissociated molecules diminishes with hilling empcialine The elkcl is Iheiefoie cqunalenl lo a continued 'ombuslion 01 '.il'lti -binning " Ol, if we icgaid I he whole thcimal MK'igy <is being devc-loped at once and then a small poilion of it is being absoibtd by the bicaUmg up of some of the molecules n consequence of (hen cneoimlcis, the effect of dissociation is ndistmgmshable from lhat of inci eased specific heat * Sou tho yocoud Koj_>oiL of tho Uutuli AflsociaUoii (Joniimltoo on Gaseous E\- l')0 ( ) CHAPTER VII GENERAL THEHMODYNAMIC RELATIONS 176. Introduction In the eailici chaptcis but JiLtlo u<-c was made of foimal mathematics in introducing the vcadci to the fundamental ideas of thennodynamics To most sludcnls Iheic is an achantagc m having these ideas so piescnled then physical significance is moie likely to be appreciated Once lhal is giasped, the btudent may pioceed to a moie mathematical heal men I with less ribk that the real meaning of the symbols uill be obscmcd m the analysis. But ti niathcmalical ticatmcnl musL be icsoilcd to if we wish to cxpiess with anything like completeness (he lelahons that hold between the vaiious piopcilics of a fluid One ol the uses to whicli these illations can be pul is in darning tables 01 charts of the piopcities of the fluid Hy then aid such tables can be compiled fiom a small mimbei of expciiinental data, and the experimental data themsihes, as well as the numbeis com- puted fiom them, can be tested foi lliermod> namic consist ency The purpose oi this cha|)tci is to show ho\\ the methods of the difleiential calculus may be applied to obtain, by mliiencc 1'ioin the Fust and ^Second Laws of Thcimodynanues, ceitain geneial iclaiions between the pwpcilics ol any fluid With some of these results the icadci ol the eaihei chapters is alieady acquainted In the next chaptu some applications of Ihe-se gnu tal lelalions to paiticnlai substances will be considered, including impelled gases, 01 ical fluids in the stale of \apoui In pailiculai il will be explained how Callcndar has emplo\ed them in calculating his tables of the piopcities of steam. 177 Functions of the State of a Fluid. Assume lhal we aic dealing with unit mass of a homogeneous (hud As was pointed out jn Ail. 75, the six quantities named theie, /*, F, T, E, f, and </>, aic all functions of the state of the fluid, that is to say their value depends only on the actual state. When the fluid passes in any manner fiom one state to anothei, each of these quantities changes en vn] GENERAL TIIERMODYNAMIC RELATIONS 267 by a definite amonnL which docs not depend on the nature of the operation by winch the change is effected, but only on whal Ihe state was bcfoic and Mhal it is after the opeialion has taken place. Tins fact is expiessed in malheniatieal lau^uaye by saying lhal the differential of any of these quanlihes is a. u peifcct" diffeicnhal. Oilier qnanlilies nnohl be added lo the list, which aic also functions of Ihe slate of Ihe fluid, such as the quantities G (01 , which is -- O) and i/r mentioned in Ail 00. In whal follows il is to be nndeislood lhal 7' means (as usual) Ihe absolute tcmpcialuu on the the rmodvnannc scale (Ail l>2) (1Q We defined the entropy </> m A.il 1 1< by the equation cty in a icversible opeialion, and the fact that cf) is a function of the slate was pioved URIC- as a consequence ot Ihe result lhal I = loi a rcyeisible cycle, a icsiill which follows from the Second Law of Thermodynamics The Second La \\ is Ihcu loie imohed in dialing (f) as a function of UK stale Ih nee Ihe lad lhal <lr[) is a peifctl difl'eiential is sometimes spoken of .is a inalhemalic.il expicssion ol Ihe Second Law II isnnpoilaiil lo notice lh.il \\hile , which is d([), is a pei led difleienli.il, <IQ itself is nol a peiled dilleicnlial, loi Ihe tnnounl of heal m\ol\ed in a change is nol a fiindion of Ihe state alone When a substance changes lionione stale lo .111- otlui, 1he amount of heal I iken in depends not simplv on wh.il Ihe l\\o sUiles <ne, bill also on Ihe nal me ol I he opeialion I >\ which Ihe change eucms Foi^lhc same icason, il \V icpiesinl the \\oik done dm inn \\ change ol stale, till is nol a pi i IV c I dilleicnlial. Since R, 1\ and J r aie- all lunclions ol the stale, il follows that the lolal heal /, Mhich is equal lo 1C + L'}'\ is also a hind ion of the stale Anel since 1 T and </> aie i also hind ions e>l Ihe slate, it follows lhal llus is also line 1 of, which is / 7'c/>, and ofj/r, A\hie'li is 1C - 7Y/> Hence ill, dL, and f/(/f, as well as r/ f />, ilK, <II\ <\V and tIT, aie pcifed differentials. 178 Relation of any one Function of the State to two others The stale of Ihe Hind (assumed to be homogeneous) is coni]iletcly sj)cci(icd when any two of the .functions of the state are known. Any third function is then determinate, lh,it is to say, it can have only one value in any paiLicular substance. Thus if any two functions (such for example as the pi assure and the 208 THERMODYNAMICS [en volume) be selected as "independent vanablcs," by icievence lo which the state is to be specified, then any I hud fiincUon (such foi example as the tempciaturc, or the total heat) may be lepioscntcd in relation to them by the famihai device of di awing a figure in which the two functions selected as independent vai tables aie i epi e- sented by lectanguLu coordinates X and F, and the third function is rcpiesented by a thud cooidiuate Z, pcipenchculai to I he plane of X and Y. This gives a solid figure, the height ol which shows, for any given state of the substance, the value of I he function Z in iclation to the values of the functions X and 1" which set NO to specify that state The surface of such a figiue may be called a thcimodynamic surface. Suppose now that the substance undeigoes an mliiulcsiinal change of state, so that the independent vanablcs change bv dX and dY lespectnely That is to say, \ve suppose X lo change to X + dX and Y to change to Y -h dY. Then Ihe tlnul luuclion changes fiom Z to Z + f/Z, by an amount </Z which may be c\- piessedlhus dZ=MdX + NdY (1), wheie M and N aic quantities depending on the iclalious ol Ihe functions to one another, and are thctcfoie also lunclioiis ol I ho state. This expression applies whethei bolh luncLions X and V \<IIN F , 01 only one of them If X vanes bnl nol F, I lion dY - and dZ = MdX sumlaily if I" N aucs bnl nol X, dX - and dZ - XtlV Hence In this notation, ( ^] means the nilc of vanalion ol Z wilh \dA. i j respect to A" when Y is constant In the language of UK calculus, ( -- v ) is the pailia! diffcienhal cocfTicicnt of Z wilh rospool lo V aJL / Y X when F is conslant, and (,} is the pailial difloicnli.il co- \dYl x efficient of Z with icspect lo F when X is conslant We might legaid the change of Z as occulting in two slops In the liist step suppose X lo change and F to keep couslanl The coriespondmg part of the change of Z is MdX, and /I/ is Ihe slope of the thermodynamic snifacc m a seclion-plane ZX. In the second step X is constant and F changes The corresponding pail of the change of Z is NdY, and N is the slope of the thermodynamic vir] GENERAL TIIERMODYNAMIC RELATIONS 269 suifacc in a section-plane ZY. The whole change of Z is the sum of thcbe I wo par Is, as expiessecl in equation (1). The slopes along the two section-planes aic expiessecl in equation (2) Combining these equations we have dZ = ( d ^} dX+(~Pl dY (3) \fl\Jjf \dYJx These equations apply when X, Y, and Z aie mtcipieted as any tin cc functions of Ihc stale of a fluid Thus, foi instance, if we think of a small change of state in which the tempeiatuic changes fiom T to T + dT, and the piessinc fiom P to P -|- dP, the consequent chance of volume \\ill be '/' Sinnlaily, il Ihc \oliimc and picssuie change, Ihe consequent change of Icinpcialuic is '<rr\ Oi again, the change ol enliopy conscc[iient on a change oi tcni- pcialme and picssuie i^ and so on ll \\ill be obvious lhal a. \ c i \ Lngc iiinnl>ci of sinnlai cfjiuilions inighl be \viilUn oul, cadi using one pan ol lunclions of Ihc sl,il( as indc pi ndi'nl \ <iiiablcs,jind c\|)icssing in I cm is of then \analion UK \aiialionol sonic Hind Innclionol the slalc These me nieic'l\ lomis ol I hi gcncial c<[iialion (JJ) Rehn mug now lo Ihc gcncial Conn in -Y, Y, and Z, suppose a small change ol slale lo occm ol such ii chaiaclei LhaL Ihc function Z undeigocs no change In thai special case dZ = 0, Ihe steps AldX and A^/rc-anccl one anolhci Coiiscqucnlly dx. when (IX and dY au so iclaled lhal theie is no va nation of Z Hence the gcncial conclusion follows that dZ\ dZ\ dY This relation between the three partial differential coefficients. THERMODYNAMICS [en. holds, in all circumstances, for any three functions of the state of any fluid. It may be expiessed in these alternative forms 'dX\ /dY\ dZ\ =- (dYj z ,dY Returning now to equation (1), dZ = MdX + NdY, the pimciples of the calculus show that when dZ w a pajcct dijfci ential, but not otheiwise, (dM\ __ fdN\ ~ In dealing with funclions which depend only on I lie actual state of the fluid the condition that dZ is a peilect diffcicnlial is satisfied and consequently equation (5) applies. We shall see umncdialelv .some of the lesults of its application 179- Energy Equations and Relations deduced from them Considei now the heat taken in when a small change of stale occuis in any fluid Calling the heat dQ uc have, by thcVnsL Law, dQ = dE + clW ( (J ) ) wheie dE is the gam of internal eneigv and rf/f is Ihc woik which the fluid does tluough mciease of its volume Since dW = PdV the equation may be wnttcn dE = dQ - PdV ( 7 ) Heie and in what follows we shall assume (hat quantities of heal me expiessecl m work units This simplifies the equations by allow- ing the factor J 01 A to be omitted. We aie concerned for the piesent only with icvcisible opcialions In any such opeiation dQ = Tcfy, hence dE = Td^ - PdV . (8)> Again, I^E + PF, by definition of 2 = dE + d (PV] - PdV + PdV + VdP + VdP . vn] GENERAL TIIERMODYNAMIC RELATIONS 271 Again, = / T<j>, by definition of *. Hence f/ = dl d(T<f>] r fif1fL \ VfJP (l^fJrh -t- t\f] f ] 1 \ = VdP - <j>dT . . (10). Again, i/j = E 7V/, by definition of iff. Hence fty - dE - d (2V/,) = Tdtf> - PdV - (Tdcf> + falT} But dE, dl, d'(,, and difj aie all perfect differentials Hence, applying Eq (5) in tuin to Eqs (8), (9), (10), and (11) we obtain at once the following four relations between paitial differential coefficients fdT\ (II) " 5) Those aic known as MaxuclTs ['out thcnnoch nanno itl.ilioiis Expiessod in woids, the lnsl one nic<ins lh<il when <lll^ ^ Hind cx:- pands ctdial)aliealh r (f/> eonsl ) I he i.itc <it which UK lempeiatme /r///v pci mill mcKcisi of \ohnnc is K[iial to the Kite at which the piessmc would i is( , pei mill inciease ol' cnliop\, if the fluid weic heated at constant \olunie The second niediis lhat \\hen a (hud is eoinpiessed aduibalicallv the Kite a I which its tempeiahne uses, pci unit inciease of piessmc, is equal to the iatc at which the vol- ume would mciCriso pci unit increase of cntiopy if the fluid were heated at constant piessuio The thud means that when a fluid is heated at constant pressure, the rate at which the volume m- ci eases with the temperature is equal to the rate at which the cntiopy would be reduced per unit increase of pressure if the fluid were compressed rso thermally The fourth means that when a fluid is heated at constant Volume the rate at which the pressure rises with the temperature is equal to the rate at winch the entropy * For the sake of symmetry (", which 18 - &, is used horo rather than G 272 THERMODYNAMICS [en. would increase with inciease of volume if the fluid were expanded isothermally The following furthei iclations are immediately deducible from Eqs. (8) to (!]). Taking Eq. (8), imagine the fluid lo be heated at constant volume. Then dV = and dE = Td(f), hence _ T til * d<f>Ji Again, imagine the fluid to expand adiabalically. Then dfy = and dE = - PdV , hence JVJ* Similaily fiom Eq (9) we obtain (dl\ ,'dl\ in) = J-> anc l r.J \dtpj j> \drj from Eq (10) (7) = V> anc ^ ( /7 ,) = fiom E, (II, (^--P.-l^.),- Collecting these results, 'ill \ . ((IK 1 80. Expressions for the Specific Heats /v',, and A",, In general the specific heats of a fluid aie no I constant, I hoy are functions of the state of the fluid We shall piocccd lo find differential expressions connecting I hem with the tcmpeiatme, volume and piessure Such expressions enable other pioperlies to be calculated when the i elation between T, V, and P is known Considei, as befoie, a small change of state during which the fluid takes in an amount of heat dQ while it expands in a leversible manner. Its entiopy accordingly increases by an amount d(f> such that Tdcj> = dQ. Its temperature changes from T to T + dT and its [i] GENERAL THERMODYNAMIC RELATIONS 273 -from Fto V + dV Take, in the fust place, tlie tempera tine id volume as the two independent vaiiablcs b> means of which ic state of the fluid is specified The change in any third quantity ay be fc> bated with reference to the changes in T and in V Thus ic hca t taken in may be written (IQ. = K v dT H- IdV . (20) -- fdQ\ ere A- V9 which is the specific heal al constant \olnme. is .... 1 ' \dTJr , , , , , fdQ\ id / is o symbol for ( 'T Since rf(2 = Td it by ISq (15) , . ^'" /a 1 -rffi ()- <1Q = K,,(IT + T ., r// r V///( l^olh sides by 2 1 , we have f/r/ = ,-." dT + I } <IV iis is ix peifcet dilfeicnlial, and thcieloie, b\ r Ivj i A '.- = ( f/ ' 7 . 7' V^/7 1 'r/A r , nee is is !> ii impoilanl piopc.i I y of A.",, To ol>l *nn a eoiiespondino'piopcilv of A r y ,, lake [lie lenipcialme 1 p res .-sine as I he Lwo independent \tuiabks and express Ihe heat ;en m ^v^lh leferenee lo I hem The. heal taken in, ilQ_, is I he same before;, beino still equal to Tdc/j We may wnlc dQ = K P dT + l'dl> (21) le K 3> , which is Ihe specific heat at constant, picssmc, is ., , ancl Z' is a symbol for ., ) f IP \(ii IT jincc 274 THERMODYNAMICS fd<f>\ But byEq (]4) (dp) T = ' Hence /' = T ( -y, and dQ = K v dT T Dividing both sides bv T, we have 21 *" \tlJL J 1> And by Eq (5), since this is a perfect diffemif ml, A\ ?J - ( d \ ( <l} '\ jTp / ~~ni I 7'/' / ( ./'/i / ^IF) T \(ll //' \(ll J j> T \ dP ) T 01 l-rl =-'!'[ 1 (J7) j thepiopeitj of /ir^coiiespoiicliny lo lli.U ol A', in Kt Fuithei, fiom Eqs. (20) and (21), 01 (/v j( - K u )dT = By wntmg dP = it ibllows LhaL K K l dV \ *- A = f ' Or by writing dV = 0, / ( 3> **- V ' ' I /r/i J V/77 , By Eq (21) or (25), eiLhci of lliese gives I Ins im,><>,l..ul rvp,. s . sion for the diffeience between the I wo spt-oilie heals, l' V 'r( (U ^ And since by Eq ( X) ( dV \ _ (M'\ f'l f> \ \dT) P (tll'Jr^ir), ' tins result may be writ leu q (M a ) rt wnbe.KW, thai AT,, um-r l,,l,ss H.an A',,. for Jj, ls ^senbally negative, ,, IC ,V,.SP ol |,K. sslm . vii] GENERAL THERMODYNAMIC RELATIONS 275 decrease of volume in any fluid, and Iheiefoie the whole expiession on the light is positive. Accoidmgly K D is always greater than K v , except in the special case when one of the factois on the nght- haiid side 11 equal to zeio, in which case K p is equal to K v This is possible in a fluid which has a temperatme of maximum density (as waLci has at about 4 C.) At the tempeiature of maximum density r , J = 0, and consequently at that point K f -K v = 0. Return now Lo Eqs. (22) and (26) In heating at constant volume dV = 0, hence by Eq. (22) In beating at constant piessure dP = 0, hence by Eq (20) K '- T ^) P < 80 > In a.n achabatic opeiation fZ</> = 0, hence by Eq (22) K t , ,dl\ _ _ tdP undbvEq (20) T j) dp),rC!r)p (32) Fin I her, by Eq (I h) ,* K,fdV\ K,,fdT\ K\<1P)* T A, fdl r \ ,dl This is Lhc lalio usually called y. Thus in I he idiabatie expansion of any fluid the slope of the IT IIIK is y limes its slope in isotheimal expansion, (IP dP 181 Further deductions from the Equations for E and / By Eq (7) dE = dQ- PdV Hence by Eq (20) dE = K v dT + UV - PdV = K v dT +(l-P) dV. In lieatmg at constant volume dV = 0, hence dE ] =K (84). ,7/71 / "- v ' ' dl J v 270 THERMODYNAMICS In isotheimal expansion dT = 0, hence, using Eq (21), dT ' P We may theiefore write -P Again, by Eq. (9) dl = dQ + VdP. Hence by Eq (24) dl --= K v dT + I'dP -I- VdP = K v dT + (l r + V] dP. In heating at constant picssuie dP = 0, hence dT -K ~ J> In isotheimal compiession (IT = 0, hence, using Eq (25), ( i T i_ T/ _ v T I \ 77, I = I + r ~ t ' JL I I dPJ T \drjj, We may theiefoie dl = K v dT + I T T 10) 182 The Joule-Thomson Effect In <i Ihiolllmg pioccss dl = (Ait 72), hence, fioin Eq (89), 'dT, __ 1 SIP ] i~ K This is the "cooling cllect" in Ihc .Jouk -Thomson poioiis plug expeiiment ol Ail 19, the cooling cllc( I which Ihc \\oilving Mind of aiefiigcrating machine undcigocs in passing Ihc ( \p,insiou-\ ,il\ c (Art 110), the cooling cflcct used cuinul<ili\ t lv l)\ r Ijndc foi I In liquefaction ol gases (Ail 123) It cxpi esses Ihc fall ol Iciupci.ihuc per unit fall of ])rcssure when any fhiidsuKcisa lluolllingopt i.ihoii, during which it icccivcs no heat fioni oulsidc. From Eq. (10) it follows Hut the cooling cl'lccl vanislu s wlu n This occius in any ideal "peifcct" gas under all conditions, lh.il is to say in a. gas which exactly .satisfies the equal ion /*/' RT I3ut it also occurs in leal gases under particular condihons of tempeiaturc and pressnic A gas tested lor Ihc Joule-Thomson effect at moderate pressinc, and at vaiious tempera hues, will he found to become waimci instead of colder on passing Ihc plug- if ir J GENERAL TIIERMODYNAMIC RELATIONS 27? he temperature exceeds a ccitain value At that temperature, /Inch is called the temperature of mversron of the Joule-Thomson ffect, throttling produces no change of tcmpciatuic Above the empcraturc of inversion the effect of passing the plug is to heat the fdV\ V as , is then less I Iran and the expression for Ihc ' cooling- \dJ. lp J fleet" rs negative. Ik-low the temperature of inversion the cooling fleet is posiluc. Tlu temperalme of inversion depends to some xlcnt on llu 1 piessuie, m any one gas It chU'eis widely rn difleicnt ases In an, oxygen, carbonic acid, steam and most othu gases is so high thai the normal effecl of Ihiollling is to make the gas older, mhvdrogeii,on Iheolhei hand, the normal effected' I hrot I ling , to make lire gas warmci, lor lire lempeuilure of mvcision is \ce})tionally low, about - <S() U C 1 * In lire Lmde process it is ssenlial lh.it I he gas to be liquefied should enlei lire app.ualus I a lempeialme below its temperalme ol nneision the pioecss an be applied lo h^diogcn only In cooling I he gas beforehand to suitably low lempeialme Taking Eqs (JJS) anel (10) logelhei we ha\e 'his pioelucl, A'j, (-7 ) , is Ihe qiianliU ol heal lhal \\onlel just \dl / f ulliee lo neiihah/e Ihc Joule-Thomson cooling cllee-l ])ei uml lop in piessuie, if il \\cie supplied lo the 1 lluiel m llu> piocess ol In ol limn 1 1 ma\ coin e me nll\ be ie pie scnle'd 1>\ Ihe single s\ m be >1 II me-asiues Ihe 1 cooling clleel, pel uml elmp in piessuie l>\ luollling, as ,i (juanliU ol heal (e \piessetl m woik umls), \\lnle ...I measmes lhal e-llecl as <i change m le mpe-ialme I dr i II lollo\\s lhal il Ihe i.mge lluough which Hie piessme (alls in llnedlhng piocess is liom /' , lo P tl , Ihe ulwle- (jnanlity e)f heal lial A\e)iilel ha\e lo be supplied lo neulrali/e UK- coeilmg effect is l^ 1 p di> = I*' 1 kf^,) - ?/ | (lp > s was si a I eel m a footnote lo Ail. 121 | * Tluu wan found l>y Ols/owwla foi a juosmuo-diop fuiin 117 atinosplioios to aLnioHpheno h Jn Callorulars Hit-am Tables UK- quantity here called p la tabulated Cor Htoain uloi the heading 'VS'6 1 " (Hoc Ait, 103) \ Cf K BucUiuglmm, Bulletin of Ihc. Bureau of Standards ( Washington), vol (5, )09, p J25 278 THERMODYNAMICS [en. Since I = E + PV we may write Eq. (41) in the loim This is instructive as showing the analysis of the Joule-Thom- son. effect into two paits When an impeifcct gas 01 vapour is throttled, that pait of the effect which is measured by the first teim anses from the fact that the internal energy is not constant at any one tempeiature but depends to some extent on I he picssuie In othei woids, the first teim is due to departure fiom Joule's Law Theie is in gencial an additional pai t of the cflect, mcasuicd by the second term It is due to depailure fiom Boyle's Law, accoiding to which PV should be constant (bi constant tempeia- ture A gas may confoim to Boyle's Law at a paihculai tcmpcia- ture and still be impeifcct in thai case il will show a cooling cffccl due to the fust term alone It is only when bolh lei ins \aiush that the gas is peifect Expenments which \\ill be mentioned in the next ehaptei sho\\ fd(pr)\ that in an impeifect gas the leim --- I may l)f (ilhci V (il ' r negative or positive accoiding to the conditions of piessmc and temperature (Ait. 197) Hence that pait of I lie Joule-Thomson effect vi Inch is due to deviation fiom Bo\le\ Law \\ill mulct some conditions assist, and undci olhci conditions oppose lli.il paitol the effect which is due to deviation tiom Joule's Law The lallu pait is always a cooling effect, the foimci may be eillx i <i cooling 01 a heating effect Al the tcmpcialmo of imeision UK l\\o[),nls cancel one anothei It may help the student to undcisland Eq (M //) if we put llic physical mteipiclation of that equation in aiiollici AV,I\ Suppose 1 unit quantity of any fluid to undcigoinul diop ol pu'ssiue in passing a poious plug 01 olhci tluotthng device \\ r i i ma\ then pul dl* =- -- 1 Suppose also a quanlilv of heat p to be supplied toil fiom outside which i list prevents any change of Lempeialme Then Kq (41 a) takes the foim p = dE + d (PV), which is equivalent to saying that in the complelc piocess, Heat supplied = Incicnsc of internal Kncigy -j. Woik done by the fluid Here d (PV) is the net amount of work done by the fluid, because it is the excess of P 2 V*, which is the woik done by the fluid as it vrr] GENERAL TIIERMODYNAMIC RELATIONS 279 leaves the apparatus, over PJ 7 ^ , which is the woik spent upon the fluid as it enters the appaiatus. 183 Unresisted Expansion ___ In. the Joule-Thomson poious ping expciinicnt the fluid, in expanding from a region of constant high piessure to a region of constant lowei piessure, does some work on things external to itself, the ncl amount oC Avhich is P V P V 1 2' > * i' i This quantity is not zcio except in special cases. But in the oiiginal Joule experiment with two closed vessels (Ait 10) Llic fluid did no woik on anvthiug external to itself The expansion thcie may thciefoic be described as stuctlv un- )esif>tt'd This distinction between it and Hie Joule-Thomson mode of expansion is unpoitant Imagine the Iwo closed vessels of the Joule cxpeiimcnt to be completely impcivious lo heat, so that no heat passes oul of, 01 into, Hie fluid as a whole dining Ihc piocess Imagine also lhal heal ma^ pass fice.lv fiom Ihc fluid in one vessel to the lluid in the othei tlnough the o])enmg between them, so that aflei expansion T becomes Ihc same in both as \vcll as P Under Ihcse conditions I he internal encigy E of Hie lluid as a \\holc is not allticd by the expansion, loi no heal is taken in or given out, and no vunlc is done This is line ol any Hind The chaiac kiishe, theiefoie, of such expansion is that 1<] is unchanged, |usl as the chaiacU nslie of the Joule-Thomson expansion is thai / ib unchanged In the uniesisled Joule expansion e.ich \essel nia\ ol couisc be ol any si/e Think of I he second \essel, mlo \\hich Ihc fluid e x- pands, as eonsisling of a gioup ol \e'i\ small chambtis which <uc suc i eessi\ ely opened, so I hat Ihe \olume ol Ihc lluid meieases \i\ sleps, each d]' \Ve slill suppose UK lempcialme ol the lluid lo allain e([iiilibi mm al each slep, and net heal to come in fie>m oul- side Then (01 each sle|) (IE = With infinitesimal steps the piocess becomes continuous The cooling c fleet in this imagmaiy process is not identical with the cooling eflccl in Ihe Joule-Thomson experiment In this pioccsb it is ( .) , namely the late 1 at \(lr ; i? which the lempeiatuic falls \\ilh mciease of volume, under Ihe condition that E is constant By Eq (30), wntmg (IE = 0, 280 THERMODYNAMICS [en and this, along with Eq (35) gives K f dT ] -T( C!P \ p-( dE \ ~ R '- 1 " Eq (4-2) expiesscs the coohnq eflecl in this imas>imuy pioccss as a fall of temperatmc, pci unit incieasc of volume, Eq (43) expresses it as a quantity oi heat, pci unit incieasc of volume, namely the quantity that would have to be supplied fiom outside to nentialize the change of tempciatuic caused by the expansion We may call this quantity of heat <r Hence in nnicsisted expansion fiom any volume V L to any volume F ' B , unclei adiathermal conditions (Joule's expansion with \essels made peifectly unpeivious lo heat), I he \\holc quantity of heat that would have to be supplied to neuliah/e the cooling effect is, foi anj fluid, / A fiuthei intciestmg lelaLion follows. By Eq (28), we had K K = T - /V 11 ^ I ~ J- \ ,/,/T , But by Eq (35), AUo.bj.EnW. On substituting these values, Eq (2) lakes the ue\v lot in A' -A' _(Z + -M F +/>) nn JV n ^ D ~~ T ~ This, hkc all the iclations i\en in the pic-sent eha])lcv, is line of any fluid. We shall letuin to it latei in connection with unpeifect gases (Ait 19-t) 184 Slopes of Lines in the /c/>, T(f>, and //* charts, for any Fluid. The slope of=any constant-pi essurc line m the 7c/> chtnL is equal to the ahsolutc tempciatuic, foi, by Eq. (1(5), It follows that all constant-piessmc lines in that chait have the same slope at points whcie they cross airy one line of constant tempeiaturc, vnj GENERAL TIIERMODYNAMIC RELATIONS 281 To find an expiession for ( - ) , which is the slope of a constant- \f/0/2' tcmpciature line in the 70 chait, we shall piocccd by a process of substitution \\hich may be followed in finding othei paiLial differential coefficients. It will serve as an example of a gcncial method Starting with Eq (9) dl = Td<f> -l- J'clP, we shall eliminate dP by substituting foi it an cxpicssion in tcims of dfi and dT, got by applying Ihe gcncial iclaLiou of Eq (I), namely, This subslitution gives ''-[ Ileucc, wilting dT = 0, '(II i v /, ,N SM.CC, b, JXI (1 I), T Sinulailv, to dud an t\[)icssioii loi f ) , which is I he slope of d(p i <i conslanl-\()luni( lint in llu f<p eh, ul, we stall horn the same (qualion loi f//, bill eliininale <U* bv substituting an t'\pu ssion I'oi il in Icims of dcp and dJ'", namely This siibslilut ion ives *' til -- T -|- J Fence, willing dl r = 0, flf\ = T -i- V fl<f> / 1 asT - J '(f^) ii ( I0fl ) Ocj (12), 282 THERMODYNAMICS fen Turning next to the T<j> chait, the slope of a constant- volume line is given by Eq. (29), dT\ T arid the slope of a constant-pi essiue line by Eq (30), fdT\ T (df) P ~~K~ f To find the slope of a line of constant total heat ( . . ) we may 1 \drf>J[ J again apply the method of substitution. Stai Lino with the equation Td<j> = dl - VdP, substitute for dP an expiession in dT and. ell (Eq (I)), dP - f flP } dT + ( dP ] dl (IF ~ + ai d ml ni i 1 -. ~rr [til \ lr __ /uZ \ ,. This gives 7rf0 =\l-V ( Tli ] J dl - J ( dT )dT, fioni which, writing dl = 0, dT\ But by Eq (40), Also, since (dV\ (dV\ (dl\ dl \~FTi = 7r T,r, il \d1Jp \(IlJp\flTjp ^/7V/> we ma> put this result m the Loim 'dT\ T r l In the IP chait the slope of an achabatic, or lino of constant entiopj, is given by Eq (17), fiom which it follows that all adiabatics luive the same slope at points wheie they cioss any one line of constant volume The slope of a line of constant tcmpeuiLme is given by Eq (38), VIT] GENERAL TIIERMODYNAMIC RELATIONS 283 To find cxpiessions foi the slope of a line of constant volume (dl\ I ,/p ) > we may pioceed thus- \llL ] y dl = dE + d (PV) = dE + VdP + PdV. - (w) . By Eq. (Jil) tins may be wnLlcn, , -"-*(). rwoothei cxpiessions which uc sometimes useful m,iv coin cm- ently l)e o.ven heio, one To. t^} ; ,.ul one foi dV ' r --"'-'i 185 Application to a Mixture of Liquid and Vapour in Equilibrium Clapeyron's Equation Change of Phase Eqiialion (30) is applicable uol onlv lo liomoociieoiis fluids, buL Lo a mivlme of l\vo phases of Ihe same snbslance, in ((]iiilibnum wilh each olhei and Iheiefore bolh aL I he s.nuc picssuic and Lhc s.unc lempd.iline 7 and V uc Ihen Lo be teckoned foi Ihe mjxUuc as a whole Say loi mslancc IhaL Lhc snbslanec is a mixluic, paiL h(]iiid and pail saLmated va|)onr Suppose Lhc ])iopoiLion of liquid Lo vapom Lo be changed by vapormno some of Lhc liquid paiL at constanL pressure, and Lhciefoic also at consUml Lcmpcratme. During) lhal piocess f-^\ is constant, for the volume of Lhc 284 THERMODYNAMICS [ CII . mixture as a whole me] cases in proportion to the heat taken in. Instead of (- r m equation (50) we may thciefoie wnlc J p I - I w ( i_ /-\i _ _ . -- V V V V ' ' ~ f w ' -. ' iw whcie the suffixes s and re iclatc to the two stales, when all is vapom and all is liquid respectively Fuilhci, the condition that $ is con- dP stant may be chopped m wilting the coelficienl wliic-h is no longer a pmhal diffeiential coefficient. Since the vapour piesent dl* in the mixtiuc is ah\avs satmatcd, P is a fund urn of T only, is simply the rate at which the piessme of sal million uses with the bempeiatmc While the mixLme is vapoii/,mg 01 condensing under vanable piessme it makes no diffeience in llu iclalum of P to T whcthci the pioecss is conducted \\ilh </> = constant, ot with V ~ constant, 01 m any othci way dining Uul pioccss fdP\ fdP\ . (IP ,. . ( -iTiJ 01 7>.J IS tlw same as ., Ilcnct wlien applied lo ,m \flTJ ^ \dT)i dl ' ' equihbiinm mixtiuc of liquid and \apom, 01 o( any I wo [Aliases Eq (50) may be mitten in the foiin 01 V - V = LdT Ul ' s ' w 'f ( Hi This is Clapeyion's Equation, which was ainved <il m \il <)S in anothci -\\ay The same lesult may be got from Eq (21) ,dV) T During vaponzation at constant tempcralme ( ) is eonstanl \al /'/' and its value is -- . Hence, chopping (lu snllix /' loi the ' s ~~ V w reason just given, we have as before FS _ FW = ^ This icsult may be extended lo any reversible change of phase which a substance undeigocs at constant pressure Dining any such change the two phases of the substance aie in eqmhbuum with one another and the tcmpeiatme is constant. Wilting A for vi rl GENERAL TIIERMODYNAMIC RELATIONS 285 I ho heat I, ikon in dm ing llic change of phase, and ("'and V" foi Ihc volumes of Ihc fusl and second phases icspcclivety (as in Ait 99), we have \ (IT V" _ //' - Cn/7 1 ) ' ' TC/P ' ^ '' Sunilaily, the cxpussion for ( ) in Eq. ( 1<7 b), namely \9// /v/r\ = 2L_ TZ ( (W \ ttyJrK, V \dl)i.' may 1)C adapted lo a mixture of liquid and \apoui in equilibimm, during the change of phase which occuis in \apoi i/ahon at con- stanL prcssuie (and temperature). In llus pioccss K,, is infinite, foi heat is taken in without use of tcnvpciatmc, and alse> dV The equation Lhcicfoic takes the foim ' idT\ T- V I' f - '" (52) V L ^ ! This applies al an\ sLage in Ihc process of \apoi i/almn, V being the volume of Ihc nuxhiic <il lhal slagc, nameh q]\ I- (1 q] f',,,, \\Iioic q is the fiaelion I hat has been vapoii/cd ( Vil 71) It gi\e-, Ihc slope ol a line ol constant lot.il beat in UK \vc I icgion (Ihc legion willun Hie boundai\ eiirxi) ol the 7V/> ch.ul A still moie diiecl means ol getting Clapu loifs E<pialion is lo use the function (1, \\liich is 7V/> - / 01 (, HvEq (10) III an\ change ol phase \\ Inch oeeuis al conslanl lempeiatiue and const, ml picssmc, such as the con\ eision ol walci into steam at conslanl pussiue, (IT and <ll' ate bolh /eio Ilenec in such a change U is conslanl, as was pointed oul m Ail 90, whuc Ihis piopeih of (i was luincd lo accoiml C'ompaic now the slate ol anv sub.slance at the beginning and end o[ a change of phase, dm ing which G 1 is const, mt Use the sullix 10 foi Ihe (irsl stale (sav uatei), tmtl Ihc sullix ,s- lor I he second state (sav sleam) ( < si **>* N ljr f - t^Mi V JJ ;> UUr, = (10 w , <^ S 6/T - V a dP == <f> w dT - r w dP * Uflod by Jonkm and Pyo (Phil. Trans A, vol r>3i, p. 300) ni collecting the Tfj> chart foi carbonic acid 286 THERMODYNAMICS [cir. dT Therefoie F s - V w = (<f, 8 - <,) ^ . But cf} s <f) lo 7 = l . Hence this again give;, Clapcyion's Equation, v -v = LdT s w TdP 1 86 Compressibility and Elasticity of a Fluid. Lcl a fluid be subjected to an mciease of pressure dP, with the lesiilt lhal Hie (.IV volume is icduced fiom V to V dV Then measuiob the volume strain, and the ratio of this stiain to dP mcasmcs the com- pressibility. The reciprocal of the compiessibihty 01 Vi y^J measiucs what \Ur / is called the elasticity of the fluid Its value will obviously depend on the cncumstances imdei which the compiession lakes place. We may for instance keep the tempeiatme constant dining ihc compiession. In that case the expression ioi the elasticity becomes - V ( 7rr I This is called the isotheimal elasticity oi a ilmd. and \dV) T J will be denoted heie by e t Oi we may prc\ r enl any hcaL fiom leaving or enteimg the fluid dining the compicssion In that case 'dP\ the expiession becomes -T (-,_.) . This, which is called Hie \dVJ {lt adiabatic elasticity of a mud, will be denoted hcie by c lt , \V'e ha\ e accordmglj' the two elasticities T-T Hence e * = d byEq (33). That is to say, the ratio of the adiabatic to the isotlu r- mal elasticity is equal to y, the ratio of the specific heats Since K is greatei than K v (Ait. ISO) e (j) is gieater than e t 187 Collected Results. All the foiegomg relations arc true of any fluid Befoie pioceedmg to apply them (in the next chapter) 'iij GENERAL THERMODYNAMIC RELATIONS 287 o particulai fluids, it will be useful to collect them here for con- Tmcnce of reibiencc dE = Td<j> - PdV . (8), I^E + PV, dl = Td<f> + VdP (9), ^ = I~T(/>, d = VdP - <(>dT (10), /; = E - T<[> , dijs = - PdV - frlT 'dT\ dr)*~ l ' u/ (12) ' dT dl\ (JV } - - f" r } di] flTJr [ '>> ( )} ^)r <"-) (15) ' (ir '"' (10), dPj T (1?) ' (23), (20), ( dK *\ = _ T ( d ^\ , . (27) V ,1T> I \,1T2I ' \^' J> \ tlJL ) ' \CIJ J p 288 HERMODYNAMICS [en s '- r &}, (29), *. -($ (30), !TW),r ~(df) r (3D, K D fdT\ (dV\ T (dP)i~ (cirjp (32), _ K v _ fdV\ fdP\ (33), /dP> fdP\ \w)r 7 \dv) T (33 a), fdE\ \dr} v ~ K " (31), P = K 1> dV) _ dPJ T \ dP T dTl (35), dP dl = K p dT+ \V-T(~\ dP . (30), dp dPj T / = -l^J_-l ,-'} ( 4] ) J .. - (42), ra] GENERAL TIIERMODYNAMIC RELATIONS 280 (1,5) P ( >' f// , rr , Tr = T + V =T -V (10 , = _ = _ ' ' A-, v\di) L y 1} ' MS) \>"h In a icvfisiblc change ol phase at consLmL j)icssiuc r -r - LllT ' s ' 10 rji I p id (; = C;,,,, or 7V/j, - /, = Trf> w ~ /, (5,3) Tlit isolhcinial <iiul adujjjalic (55), (56) II T CHAPTEE VUI APPLICATIONS TO PARTICULAR FLUIDS 188 Characteristic Equation The genual Ihermodynamic iclations consideied in Chaplci VII can bo applied lo dclcimnic I he piopeities of a paiticulai (hud wlicn an equation connecting one of its pioperlies with t\\o othcis is known An equation ol Ihis kind is, called Lhe "Chaiactenstic Equal ion" or "Equal ion ol State" loi the given fluid. It is based upon expu unc uLil know- ledge of how the numerical \aluts ol sonic out pioptih, such as the volume, depend upon those ol hvo olhci pioptilies, sueh <is the piesbiue and the tempeiatuic, Ihcse two being used ,is inde- pendent \anablcs loi specif} ing Ihc stale The 1 most usual loim ol characteristic equation is one connecting l r with /' and T Sueh an equation, when it can be established, is ol lund.inu ulal iiupoi- tance m the calculation of oLhei piopeilies Jiul lakcu by ilscll il cloeb not allow all the thcimodynanne quaulilies lo be <h leinimed foi that puiposc it must be supple menletl by dal.i ugaiduig Ihe specific heat, 01 (what conies to the same Hung) l>\ dal.i ,is lo Hit lelation of the internal energy lo the temptiahne 189 Characteristic Equation of a Perfect Gas The simplest case to considei is that of an ideal g.is coid'oniiing exacllv lo llu equation j>y / t >y where R is a constant and T is Ihe absolute lempeialuie on Ihe theimodynamif scale We discussed some of Ihe piopei lies oi sueh agas m Chaptei I, but iL will be mstiuelive now, as a firs I example of the method, to show IIOAV certain results which weie obtained there follow dncclly when this ehaiaelenshe equation is mlcr- preted by applying to it some of Ihc geneial ieli lions oi Art 187, which hold for all fluids By diffeientiatmg the characteristic equation ol the ideal gas, we have ir. vm] APPLICATIONS TO PARTICULAR FLUIDS 291 [ence m such a >as, P - dT) v ~ V~T' \dT)p~P~T } d z P\ _ " y Eqs. (23) mid (27) of Chap. VII, m any fluid, dK,\ _ T (d*P\ fdK,\ r dr ) T - L (dT*) y dlld I dP ] r = ~ J ence in the ideal >as, [uis it follows fiom the chaiacteristic equation that both K ' and 3, aie eonslanl at any one tempeiatme, m olhei \\oids they aie dependent of the piessmc They mav ho^e\ ei \aiy vullitcmpeia- le Ihe chaiaeLeiishc equation yi\es no mfoimalion on that )int By Eq (28) ol Chap VII, m any llmd. ./V. JJ -tY. y J. Liicc m the ideal <>as, z r This apices willi Yit 20 The lacloi --/ is onultcd because quan- les ol heal aie hrie expiesscd in \voik mills (-Vil 179) Hy Kq (10), Chap VII, in any llmd Ihe cooling effect in the ule-Thoinson poious |)lu^ evpeiunenl js 7' ('"" V' V the ideal ya.s ( "' ) = -, , hence the quantity m square brackets -pdv. and theic is no eoolmo cffccl. By Eq (30), Chap. VII, m any fluid, the ideal gas T [ Tr =, ) = P, hence \aJL j Y dE = K v dT, 292 THERMODYNAMICS |c" and since K is independent of the pressuie it follows that the internal eneigy of the ideal gas depends upon I he tempci aline alone. By Eq. (39), Chap. VII, in any Quid, In the ideal gas T = V > hcuce and since K v is independent oi Ihc picssuie it iollows that the total heat of the ideal gas also depends upon I he Unipeiatnrc alone These icsnlts show that a gas which conlbims exactly to the chaiacteiisiic equation PV = RT (T bcino Ihe lempcialine on the theimodyiicinnc scale) conforms exactly bolh lo Boyle's Law (PV constant foi any one tempeiatuie) and lo Jonle\ Law (E a function of the tempeiatuie alone). It is therefoic ''pi-ilccl " in Ihe sense of Art. 19. When the equation PV = RT was inliodueed in Ail IS Ihc s} mbol T denoted tempeiatuie on Ihe scale oL Ihe gas Iheimomelei, that is to say a scale, defined by the. expansion ol Ihe gas ilsilf, and the gas was assumed to confoini e\aclly to Bo} le's Law Bnl il it also confoims exactly lo Joule's Law, Ihe scale ol Ihc <>as Ihu- mometer coincides, wilh Ihe Ihumoeh nainic scale (Ail \->) 190 Isothermal and Adiabatic Expansion of Ideal Gas In the ideal gas, since E depends upon Ihe lempeialme alone , it is constant dining isothcuual expansion, and theieioie (he \\oi k done by the gas is equal to the heat it lakes in The picssme. \aiies in- versely as the \olunie By Eq (33 a), Chap VII, foi the adiabalic expansion ol any llincl, 'dP Hence in the ideal gas So that in the adiabalie expansion of an ideal gas, dP cW_ p- + y v - If now we make the fuither assumption that y is constant, which en] APPLICATIONS TO PARTICULAR FLUIDS 293 equivalent Lo assuming thai the specific heal docs not vaiy with mpciaturc, this gives on integration ISL ^ J + 7 ly<. 1' consLanb, PV = cons Ian L, hich is Lhc adiabatic equation of a peifecl gas with constant >ecific heat, ai lived at otherwise in Ait 25. 191. Entropy, Energy, and Total Heat of Ideal Gas By qs (8) and (9), Chap. VII, in any fluid, ,, _dE + PdV _ dl - VdP fl( P ~ 2' ~ T i the ideal gas dE = K,,dT, dl = K v d r l\ , P I? id since ^ = , . . rr dT ,, dr (IJ> - A,, I- A' P cncc \f\\e aqain assume thai Ihc specific heiit docs not vai^ \vilh c tempeiatuic, E - K,,T | cons I an I, / --- A ,,T -| constanl, r/ = K l loo, T + li loq t V -|- conh-lanl A',, loq, 7 1 /? loo f 7-* + eonsl.iul The values of the constants depend on \\hat inilial stntc js chosen I he slaitmo-poinl of the leekomnq. H is only changes in E, /, id r/> I hat can lie determined by Ihese fommlas. 192. Ratio of Specific Heats Method of inferring y in Gases om the Observed Velocity of Sound We saw (Ait 18G) lhat any fluid the latio y of the two specific heats, Kj,/K v , is equal the lalio of the adiabalic elasticity c tj> lo the isothcimal elaslicity Also that /f jn-, "<--aH, cncc in a gas foi which PV RT, c t - V =- P, and Cll> - yP 294 THERMODYNAMICS ICIT This i elation has been used as a means of finding y cxpen- inentally m an and othei gases which ab ordinal y temperotuics and pressuies very neaily conform to the equation PV = RT The method is based on Newton's theory of the transmission oi waves of sound Newton showed that waves of compicssion and dilatation, such as those of sound, tiavel thiough any homogeneous fluid with a velocity which may be expiessed a^VeV, wheie V is as usual the volume of the fluid pei unit mass (the leciprocal of the aveiage density) and e is the elasticity, in kinetic units. It was afterwards pointed out by Laplace that in applying this result to the passage of sound through an or othei gases e should be taken as the adiabatic elasticity e (]l , for the compiessions and dilatations follow one another so fast as to leave no time for any substantial tiaiisfer of heat fiom the poitions that aie momentarily heated by compiession to those that are momentanly cooled by expansion. Hence in an under atmospheric conditions, or in any othei neaily pei feet gas, sound tiavels at a late equal toVyPV This fact is used as a means of cleteiminmg y by measuimg the velocity of sound 01 (what comes to the same thing) bv measuimg the wave- length in sound of a known pitch In an at C and a picssuie of one atmosphcie the \ allies gi\ en by \aiious obseiveis for the \elocitv ol sound lange (join 33,000 to 33,240 centmietics per second ' Under these conditions the volume of one giamme of an is 773 1 cubic cms , and P is 10133 y 10 dynes pei sq cm (Ait 12) Hence, takmo an avei age of 33,150 for the velocity, 33,150 - \/y x 1 0133 - 10'- > 773 1. which gn es y = 1 103 193 Measurement of y by Adiabatic Expansion Method of Clement and Desormes. Anothci method of dcteimuung the value of y in a gas is bv an expeiimcnt due originally to Clemen I and Desoimcs and impioved on bv Gay-Lussac and olheis A quantitv of the gas is contained in a large vessel at a picssuie sonu - what higher than that of the atmosphcie, and at almosphei ic tempeiatme Theie is a pressure-gauge attached, and a lap which may be opened to allow some of the gas to escape quieklv On opening the tap, the piessure falls suddenly to that of the almo- spheie when this happens the tap is at once closed Then the * See Raylcigh's Tlieoiy oj ftuiind, vol ir mj APPLICATIONS TO PARTICULAR FLUIDS 295 iicssmc of the gas that icmains m the vessel slowly uses, because he tcmpeiahne, which had been reduced by the sudden expansion f the gas in the vessel while the tap was open, rises gradually to he value which it had at fhst, namely the tempciature of Lhc uiroundmg atmosphcie. When this piocess is complete the final ircssiuc is noted. Let the oiigmal picssure be P l5 the picssure of he atmosphcic P., and the final picssuie P 3 The change fiom \ lo PI is appioximalcly adiabaLic on account of its suddenness . he change from P 2 to Pj occuis at constant volume Let F 19 V\ nd F 3 be the volumes ol the gas pet umt mass, at the Ihicc coiic- ponding stages. Then F 2 -= F 3 . We have, in the ndiabatic ex- lansion, p T/ y _ p v y L i' l r * 2' 2 > nd since the initial and final tcmpciatuies arc the same, . P, 7'o Iciu>c 2 l /allies ol'y me accoidmglv I'ound b\ obscn ing these thue presumes Oxpeiiments by Lumnici and Pimgsheim, using llus method in n impioved fonn, oi\c- 1 J025 as the \aluc ol y loi noinidl an An aihei application of Ihc method by Ronlgcn ga\ e 1 105 ' 194 Effect of Imperfection of the Gas on the Ratio of .pecific Heats II has been aluadv mentioned that in a peilecl halomic gas Ihe lalio y, as deduced fiom the moleculai theoiy see Appendix II) 5 shoulcl not exceed 1 I In an the lalio, aecouling o all the evidence, is, at oidmaiy lempcialuies and piessuie-s, hghtlv gicaUi This is due paitly lo the presence- ol about me pei cent of (monalomic) aigon, but mainly lo I he facl that jr is ,in impei led gas, deviating to a small extent boll) fiom ioyle's Law and fiom Joule's Law H> K<i (It), Chap VII, in any fluid, r r _ A j, A ,, - y, vhere p is the cooling- cflccl m the Joule-Thomson poious plug xpcnment (Ait 182), and a is the cooling effect that would be * Seo Plosion's Tlicmii of Heal, Chap TV 296 THERMODYNAMICS [c H found m unresisted expansion (Ait 183), without gmn 01 loss of heat in either case. In a perfect gas p and a aie both nil, and Ihc expression on the light becomes PF/T, as it should Willi air (nndei usual conditions) both p and a aie small positive qnan Lilies p was measured in the Joule-Thomson expeiiments, and a, though iL has not been dnectlv measuiecl, can be mfeiied fiom known cxpcii- menbal da ta Hence K^ K is a little gi cater than PV/T, which is the value it would have in a perfect gas The ratio y is also a little greatei in noimal an than it would be in a peifecl gas. In any fluid (P + a) (V + p} y= 1 + K,T In an at ordmaiy tempei atiu cs the impcii'cction inci cases (P + cr) (V + p) moie than it increases K v and consequently makes y slightly exceed the ideal value 1 4 But at high tcmpcia- tures K v is much increased (because the molecules then acqunc energy of vibiation) and y is substantially i educed 195 Relation of the Cooling Effects to the Coefficients of Expansion The expressions for p and a given in Eqs (<J ] ) and (4<3) of Chap VII may be put in anothei foim Mhich is coin enienl in dealing with impeilcct gases By these equations, in any fluid, ' Heie ( -j=A may be wntten aV, wheie a is the fiactional mcicase \al >p of volume pel degiee. on the theimodjnamic scale, when Ihc Hind is heated at constant piessure. Measuiecl at C a is the co- efficient of expansion at constant piessuic, or what is sometimes called the ' L \olume-coelTiciciit." Similaily (-7) niay be wutten /3P, wheic j3 is the fiactional \(LJ. / f increase of pressure per degree on the thcrmodynamic scale, when the fluid is heated at constant volume Mcasmcd at C. j3 is A\hal is called the "piessmc-cocm'cient." Hence at C. F + Po = 273 la Q V Q> and P + cr, --= 273 lj8 P , the suffix being mtioduced to show that the quantities conceined aie all to be taken as at C. The results of the Joule-Thomson porous plug experiments may inj APPLICATIONS TO PARTICULAR FLUIDS 297 3e used to calculabe p Thev showed thai with an the cooling 'ffcct of passing the plug was nearly pioportional to the drop in jiessure It was diffeient foi different initial tcmperalmes, bc- 'oming less when the initial tempeiatuie was raised. With air at ) C the cooling effecL (according to the foimula in Art. 123) was ) 275 for a pressuic-chop of one almosphcu in passing the ping -lence, using c o.s units, foi air at C. we should have '(IT\ _ 275 J 7 ~l 0133 x 10 kVe may Lake 7v,, as 0211 caloiy (Ait. 161) equivalent in c c, s nuts of woik Lo 211 x 4 1868 x 10 7 . Multiplying the values of fdT K v and . ) we obtain This is in cubic ccnhmelics pci giammc, the dimensions of p DC'ing Ihc s<unc' as Ihoso of V, namely woik \olume pussine > mass mass We ma\ appl\ I Ins lesnlt ol Ihc poious plug expeinnenl to '.ilcul.iU the eoellicKiil of expansion \\hcn an, at C , is heated indtr a conslanl picssinc ol one dtmospheie Ihiounli one degiee >f I he Ihcimoch'iiainic scale We had - r o I- Po 11 " 27.5 1 T In an aL C. and a piessuie ol one atmospheic, Ihc \ ohimc ol one gamine it, 773 5 cub cms Hence under llusc condiLions \\c should l] ^ e 7735-1-271, "* - 273 1 , 773 5 = 0(W 75 1'his is shi>lilly laigci than the. mean coclliciciil lhal is lonncl \\hcn he expansion ol an at a consLanl piessmc ol one atmosphere is ncasuicd over a i<ine of IcmpciaUire Irom (\ lo 100 C Again, taking I he iclalion lvalue of O-Q can l)e infcricd when thepicssiue-cocHicient is known. If j8 foi au- be taken as about 003G71, 7 J + cr n becomes 1 003 lP , making CT O - 003 \P In a perfect gas both coefficients, and /3 , would be equal to ~, or 003G617 The scale of the perfect-gas thcimomclei, itO 1 208 THERMODYNAMICS LC-II. whethei of the constant-volume 01 constant-piessmc type, svould coincide at all points with the thermodynamic scale* 196. Forms of Isothermals. Diagrams of P and J", and of PV and P. Taking any ideal gas, which satisfies the cliai aclenslic equation PV = ET, leb us diaw its "isotheinials" on a diagram whose cooidmates aie the volume and I he prcssuic The chaiactci - istic equation shows that these curves ait leelanguLTi hypeibol.is Volume Fig 84 Piessuie Volume Isothcimals fm a Poifoct Gas (ng. 84), ioi while any tempeiatuie lemains consl.iul P v.uus m- veisely as V These isotheimals foi an ideal gas slioiild be t-oni- paied with those ioi a liquid and its vnpoui ahead v illush.ilc-d in fig 14 (\it. 76). to which we shall lecui prescnlly. Anothei kind of isothermal cuive, which Ama,il showed lo he useful in dealing with ieal gases, is duuvn hv Liiloiii> as cooulmnlc s the pioduct PV and Lhe piessine When this nu Ihod is applied lo an ideal gas the isolhcimals are simply hon/oulal slum.])! Jmes (fig S5),smceat anytemperatuicPI'is consLanl This is.n'ohx inns * Refeience should be made to Callondai's papm "On tlu^ ThnnHxlynnnnu,! U l re B tionoftlioGasTheimometer"(P7u/ Hag 3>m 1)OU) ], na a, , o.n.i ,,| ], the absolute zero may be deteimmed and mteivals on gas and tlu-iiumlyn a muHi.il.s compared, by making me ot the Joulo Tliomson cnolinjr ,floU ami UK- moiiH,,,, (I coefficients of expansion. vin] APPLICATIONS TO PARTICULAR FLUIDS 299 tesb of whcthei a gas obeys Boole's Law If it docs, then the iso- theimals of PV in i elation to P will be horizontal straight Imos 200 C 100C CL o - 1 00 C Fig 8.3 Amagaf, Isotlici mala lor a Petted Gas wlulhci I he gas also obeys Joule's Law 01 no Any cmvalmc in these lines, 01 any deviation 1'ioin Hie hoi izonLal, means adipaituu I loin Bo\ le's Law. 197 Imperfect Gases Amagat's Isothermals of PV and P No leal gas confoims slncllv to Boyle's La\\ The cxpeiinienls of Pressure Fjg 80 Typical Amagat Isothoi mal foi an Impel foot Gas Andrews, Amagat and others have shown lhat the depaituic fioiu Boyle's Law becomes moic and moic marked as the entical point 300 THERMODYNAMICS CTI is appioached Amagat'.s expenments on the compicssibilily <>!' oases, which extended up to veiy high picsMiies, show lhal Avhcn a line is diawn to exhibit the lelation of PV to P at u conslaul tenipeiature, its geneial form is that illustiatecl in fig 80 InsLcad of being a houzontal stiaiglit line, as Boyle's Law Mould require, in am 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 Fig 87 Ainagal's Isotheiimils fo? Cailnmic A( id it consists as a rule of two neaily stiaighl pai Is, A and (', one 1 sloping down and the othei sloping up, milled !>v a snioolh cum Theie it> consequently on each isothernuil a minimum value oi P) r at a paiticular pressure Foi picssmcs less than Llns 7.3 j is negative, foi greater pressures it i-> positive The particulai vin] APPLICATIONS TO PARTICULAR FLUIDS 301 pressure at which the minimum of PV is found depends on the tempera! me With using temperatine the position of the minimum point B shifts fiist to bhe right and then to the left, and if the 46 44 ^ 42 ^ <a - 40 S Q. 38 36-.. __ 34 32 30 "" ----"' Pitsstire in aim 26 ' ' 20 40 00 80 100 120 140 160 180 200 220 240 260 280 300 320 Pig SS A. in fij.ru I 's I^uthoimals fui Nitrogen 44 ^ Q- 42 _ 40 -S qp oo 28'-- Pressure in aim 26 ' ' 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 Fig 89 Amagat's Isothei mnls foi Ilydiugon temperatuie is high enough it may leach the axis of zeio piessuie and disappear, with the result that the whole isothermal then consists of an upwaid-slopmg line like BC. The geneial features of these isotheimals will be appaient from 302 THERMODYNAMICS [en. figs. 87, 88 and 89, which aie lepiesentatrve examples ot Amagat's curves* The tempeiature for which each i&otheimal is diawn is 14' 4-100 13 T2| 11 ' Pressure in atm 20 40 60 80 100 Fig 90 Witkowski s Isotheimals foi Au 120 maiked on it In fig. 87, which relates to caibomc acid, Lhe tcni- peiatmes foi which the selected cmves are diawn aie all above Lhe cutical point, but the lowest is not fai fiom it The left-hand * E H Amagat, Annalcs de Chmue el de Physique, vol. xxn, 1881 See also vol xxrx, 1893 in] APPLICATIONS TO PARTICULAR FLUIDS 303 ranch of the curve consequently slopes down veiy fast, if an othermal were drawn for the critical tempeiature its dnection at ic cutical pressure would become vertical, foi at the cutical point - ' ) is infinite. At highei temperatures the left-hand bianch dl J ' opes down less steeply, and within tlie lange of this diagiam the iiiiimum point inoveb to the light, but (as Anther experiments loved) at highei temperature-) still it moves to the left This cituic is apparent in the curves of fig. 88, which i elate to mtiogen. lieie the incasiuements weic 1 made at tempciatuics much moie imote from the cutical temperature The dowmvaid sloping lanch is shoit, and becomes shoiter Avhen the tempeiature is used Finally, witli hydiogcn (fig 89), \\heic the cntical point as even moie icniole the minimum has disappeaied, and each otheitnal is a hue sloping up along its whole couibc. At any \ci\ >w ttmpeicituic, howcvci, an isolhennal foi huliogen would ha\ c bianch sloping doumvaids, loi modciate values of P, followed bv minimum of PV and then an up'Udid slope ]ust as in olhci gases ivpeumcnls on an, b\ \Yitko\\ski" 1 , uhosc cui\es aie lepioduced i fig 90, show lh.it lot modciate pressings a PV, P isotheimal loi 11 slopes upuaids all tlic \\av at 100 C (01 o\ei), but at lou 'inpeiatmcs such as 100 I il slopes steeply do\\ mvaids lowaids minimum \alue ol PV and then uses The locus oi the muu- mmofJPr is mdicaled in Jig 90 In a dotted line. These conclusions ic in lull aceoid with the usulls that have just been slated 198 Isothermals on the Pressure-Volume Diagram An- UKI melhoel ol e \lnbi I ing I he- elepai tine- of leal gases fiom Bo% r lc\ ,<iw is lo diavv isolhennal Inus on a chaplain ol the 1} pc ol g SI-, I he cooiehnales of ^\l]l( l h aie simplv the 1 piessuie and the olume. An example ol such a diagiam \\as dc sen bed in Ait 7(J '01 convenie-nee of icJ'eience it is lepioduecd as fig 91. At any igh tenipcuiLure an isothermal (such as 6') does not differ \ciy bviously fiom <i rectangulai hyperbola, but at lower tempeiatmes exhibits a point of inflection as m F or E Below the cutical tempeiature each isotheimal consists of spaiate paits, namely the pait AB, in which the substance is a omogeneous liquid, and the part CD, m which it is a homogeneous apour. These are joined by the stiaight line BC which exhibits le change of phase from hen.nd to vapour Dining that change the * FM. Mag , April, 1896 301 THERMODYNAMICS en substance is not homogeneous; it consists oi a mixtuic of the two phases, liquid and vapour The loci of B and of C together constitul e the boundary curve, the apex of which is the cutical point. The isothermal for the critical tempeiatuie E, fig. 91, touches the boundary curve at the ciiti- cal point. Its duection at the cutical point is honzontal and it has a point of inflection theie , consequent!}' at that point 'dP\ .. , fd*P\ n , u \dV) = and T Fl S 9l 199 Continuity of Liquid and Gas The essential conti- nuity of the liquid and gaseous states in any substance will be lealized if one thinks of a piocess by which the substance may actually pass fiom one to the othei state without any abrupt change, such as that \\hich occuis in the boiling of a liquid Staitmg liom E (fig 92), wlicu- the substance is a liquid, we might heat it at constant \olunie to a tempeiatuie equal to the cutical tcmpciatuu (01 highei) This bungs it to H Then it might expand isothcimalh along I he line HI, and then be cooled at constant volume fiom / to (' Al (' il is a satin ated vapom Dining each of these steps the substance, has lemamed homogeneous, the passage fiom liquid Lo \apom has taken place in a continuous mannei it would be impossible to point to anv stage of the piocess as the stage ol transition horn one phase to the other. It is obvious that any isothermal highei than the cutical isothermal E would seive equally well loi the step in which the substance expands The idea of continuity between the liquid and gaseous states ic- ceived a icmaikable development in the speculations of James Thomson ^ He suggested that we might think of the ciuvcs AB and CD as parts of one continuous curve ABJKLCD (fig 93) The paits BJ and LC correspond to real phenomena of the kind icfciied to m Ait. 79 For in certain cncumstances the pressuie of a hqiud * Proc Roy Soc . Nov. 1871 Collated Papers, pp 270-333 m] APPLICATIONS TO PARTICULAR FLUIDS 305 Volume lay be i educed below the saturation pressure corresponding to letempeiatuie, without vapon/atiun, and a vapour may be com- icssed beyond its satu- itron pressure without i \E Diidcnsatron Points be- .vecn B and J, and etween C and L, ac- jrdmgly represent con- itions in which a homo- encons fluid may tem- oianty exist in meta- Lablc stales (compare 1 1. 1 35). But points bc- kveen J and L cannot be ;ah/ed in a homogcnc- us fluid they would be r jmp!e tcl\ unstable, foi liev would reqiine the icssuie and the \ olume > increase together lf ~ Ft nee the eounecling poilion ol lire curxe is no rnoie than a i.Uhenralieal abstraction, bill rl allows a continuous expression u /' in rclalion lo V to be rlcipicted i'or isolhc-inials clow the cnhe.il tcnipeia- uit as well as lor isolher- lals abovt that leiapcia- IIH The stiai^hl hue BC L-pie-scnls lire ouhnaiy loecss o\ xapon/alion or ondcnstilion Ml consUinl ressuie II is interesting lo otiee that the theorclieal J onnceting ciu\f, which we ray call the James Thomson Volume .rave, must satisfy this , , , ,, , Fig <J3 Jainos Tlioinaun a ideal isothermal her nrodyna nne condition, hat the aiea BJK rs equal to the area KLC Foi we may oncciVL the fluid to be taken through a complete cycle, from B luough JKL to C, and then back to B by the straight lure CB )urmg tins cycle its temperature docs not change, and thereibie, T! T 20 K 306 THERMODYNAMICS [cu by Carnot's pnnciple, the woik done in the cycle as a whole is nil Accoidmgly $PdV for the complete cycle must \ainsh. hence the positive aiea KLC must be equal to the negative area BJK. It follows that when we are able to chaw foi any fluid the theo- retical isothermal AJLD, fiom a knowledge of the chaiactenstic equation, i\e may go on to deteimme Lhe satmalion picssme corresponding to the tempeiatuie foi which the curve is calculated, since that is the piossure at which the straight line JSC must be drawn to make the aiea BJK equal to the area KLC. 200 Van der Waals' Characteristic Equation A form of chaiactenstic equation, applicable to any fluid, was devised by Van dei \VaaIs 81 which appioximately expiesses the i elation of P, V and T under all conditions of the fluid though any lange of density and tempeiatuie, fiom the state of liquid to thai of gas 01 vapoui at low piessuics, when the behaviour appioachcs that of a pcifect gas Although Van dei Waals' equation cannot be accepted as exact it gives icsults which coiiespond in a lemaikable mannei with the broad featuies that are exhibited by ical fluids, in all possible liquid or gaseous states, and tluo\\s light on the phe- nomena of the cntieal condition and on the question of conlinuih of state between liquid and gas Van dei Waals' equation was based on the kinetic thcoi v of gases Nomoie than a rough outline can bcgiv en hcieof Lhe consieleialions involved in flaming itj. The kinetic theoiv shows that a gas Avlneh consists of colliding molecules will confoim to the ideal equntion PV = RT onlv if (1) the size of the molecules is indefinitely small cornpaied with the space tia^eised by them between Iheu en- counteis, and (2) no appieciable pait of the cncig> ol Ihc gas is due to the mutual attiaction of the molecules foi one anolhu Ncithci of these conditions holds in a ical gas In a ical gas the \ olume ol the molecules is an appieciable pait of the whole \olumc oce-upied by the gas, and it is onlyaftci making a deduction Joi il lh.il wchavc the volume which can be i educed by applying moic press me Again, dining then encounteis the molecules atluicL one nnolhci acioss shoit distances so that internal woik is done in sepaiatmg them The lesult is that this attraction between the molecules assists the * The Continuity of the Liquid and Gaseous States of Mallei , published in Jhitt li in 1873, Eng Tians in Physical Society's Physical Munam,, \ ol i, pail in, 1H ( )1 f Students asking to puisue the mattei should consult Joana' Di/nuniu nl Tliny of Oases, second edition, Chap VI fm] APPLICATIONS TO PARTICULAR FLUIDS 307 Diessure exerted by the em elope in preventing the gas fiom ex- pandin If the fiist of these t\\ o effects stood alone we should have where /;, which is called the "co-volume." vepiescnts the deduction luc to the volume of the molecules But in consequence of the >econd effect we have to add to P a term depending on the attrac- 1011 between the molecules Taking any miagmaiy plane of .eparation between two poitions of the gas, the atti action between noleculcs across thai plane will depend on the number of molecules ,vhich aie at any moment so near as to be exeieising mutual foices n othei wouls upon the number of encounteis that occm on the .eparatmg plane pci unit of time But thai \\ill depend on the >quaic ol' the density, foi if is piopoitiomil to the pioduct of the nimbeis of molecules pei unit of \ohime on the tuo sides of the )lanc Accoidmgly Van dei Waals takes a/V* as the teim which is Lo be added to P lo lepiesent Ihe effect ol the mutual moleculai itliaclions lie 1 1 cats n and b as constants foi any particular fluid [Iis ehaiactciistie equation llieieloie lakes the foim P I I Numciical values of Ihe constants can be found foi any fluid by obseixing expciimenlalK ihc lelations of piessuic, \olnme ind tempeialuie in dilleient conditions of the fluid, 01 lhe\ nay be mlened fiom olhei ex[)eiimental icsulls Van elei Waals' equation is intended to appl} to any homogeneous state, gaseous 01 liquid ll docs in fact icpiodiice uilh lemaikable eom- >iehc nsi\ encss the clucl |)lunomena ol both stales, and also those jf the entical i)omt, I hough in some pai heulais it fails to gu e exact [iianhlalix e icsults II may help lowaids an appiecialion of Ihe physical meaning of he leim if we considei llic isotheimal expansion ol a gas to ivhich Van dei Waals' equation ap])lics When any fluid expands n any manner the he.il taken in, (IQ, is, by Eq ('21) of Chap VII, dQ = K v dT -I- T K\ dl r [f the expansion is isolheimal this becomes ,lf\ rit I "* \ jjr 202 308 THERMODYNAMICS [en Now in a Van dcr Waals gas ET a . ~ ~ ( a} ' f I 1 i horn which (df) r =v^l Hence in the isotheimal expansion of sneh a gas - 2 dV ..... (3). But PdV is dW, the exteinal woik done during the expansion. Compaung this with the general equation dQ. = dW + dE, we see that in a Van der Waals gas theie is an increase of internal eneigy (dE) dining isotheimal expansion which is equal to ^ dV We may legard this, as mteinal u.oil\ done against a cohesi\e foice ~^ resisting the expansion, independently of the exteinal pi essuie P. In a peilect gas there Mould be no change of E in isothcimal expansion (Art. 189) On assigning vaiious constant \alues to T the Van dei Waals equation gi\ es isotheimal ciu\ es which ha\ e all the 1 gcncial chaiac- teristjos of those shown in figs 8Gto93 When the substance is in I he gaseous condition and at any \crv low picssuic, V is so huge that the teims ajV- and b become neghgjbly small the gas then ajjpioxi- mates to the idealh peifect state and the equation oi\ es ncail\ the same lesiilts as those of the pei feet-gas equation PV RT At highei piessuies both of the niodihmg turns become impoilanl The equation may be \\ntten thus, as a cubic in V* W\ T , y aV ab This gives thiee loots, leal 01 miagmaiy, foi F, couespondmg to any assigned value of P, on any isotheimal When the tcmpeiatuie for which the isotheimal is drawn is highei than the cutical Icm- peiatme, only one of the thiee roots is ical that is to say theic is only one value of V foi each value of P on any isotheimal above the one that passes through the cutical point. For any tempera- vm] APPLICATIONS TO PARTICULAR FLUIDS 809 tuie below the cutical tcmperatme all thiee roots aie leal in the mathematical sense. The isotheimal ciuve calculated fiom the equation then takes the continuous foim conceived by James Thomson, and ilhustiated m fig 93. One of the thiee loots coiie- sponds to a point on the cuivc AJ t one to a point on LD, and the thud to a point (not icahzable cxpenmentally) on JL. Van del Waals 1 equation makes the pioduct PV, foi constant T, vaiy in the mannci indicated by Amagat's isotheimals, showing a minimum at a pailiculai value of the piessmc that depends on the tcmpciatiuc foi which the isolhcimal is chaMn Wiitmy the equa- tion in the foim RTV a PV = _ V-b V" and diffcicnliatmg with icspect to P, kctping T constant, we have d(PV)\ ( dP } T Smccon any isolhcimal - iszeioal the minimum ol PV, the quantity within UK sqiiaie bi.ukels musl \anish al Ihal pomi Ilcnce, on any isolhcimal, the minimum ol PI' is lound \\hcn Ihe is such Ihat V V a (M / __ b\*J>R r r This shows that llu \ oluiue at \\hich Ihc minimum ol PI' occms on an\ isolhcim.il becomes gicatci as Ihe lompcialine is i.usc'd In the pailuulai case \\lun the Lcmpcialuic is so high that Ihc minimum occms on the PV axis, \\hcic P is /eio, }' is indelmilely laige, 1 then becomes equal lo 1, and T is gi\cn by the equation T , ,, Ilcnce in a fluid which satisfies Van dci Waals' hit equation an Amagat isotheimal foi a Icmpeuiluic, equal to would slope upwaids along its whole couise, \\il\\ incieasing /*, but an isotherm, U for any tempcialiue IOMCI than this would fiisl clip lowaids a minimum of PV and then use 201 Critical Point according to Van der Waals' Equation To find Ihc culical point of a fluid which satisfies Van dci Waals' equation, we may mosl conveniently wiitc Ihe equal ion m the foim RT a P = V-b F 2 ' 310 THERMODYNAMICS [en t , . /dP\ - RT 20 , ... fromwkch _ + . (6). , d 2 P 2RT and At the critical point (1*\ =0 and r: T : ) =0. \dV/ T \dV~J T Hence, wilting T f P c and V c foi the cutical temperature, pressuie, and volume, we should have, in a Van dei Waals fluid, This ives from which V c = 36 (9). It follows from this icsult and fiom (8) aboA e that RT C _ 2a^ (3b^b) z ~ 27b*' fiom which T * = &Sb (10) Also, fiom the oiigmal equation, P = KT <- a = A " _ JL = a n i \ c 3b - b Qb 2 27 ft* Oi 2 27b 2 Thus if the constants a and b as well as 7? \\cie known for a qns which strictly satisfied Van dei Waals' equation, the cubical A olumc, tempeiature and piessuie might he calculated 01 coincisclv the constants might be mfeiied fiom known values of T it P c and V f It follows also that in such a gas the bhice cutical quanlitios would be connected by the relation P V = RT (1*) * c c 8 c \ * " 'a which shows how widely the condition of the fluid then diffcis fiom that of a peifect gas. In applying his equation to caibonic acid, Van dei Waals de- duced fiom the expenments of Regnaiilt and of Andunvs tlicsc values a = 00874, b = 0023, 7^ = 003085, the unit of pres- sure being one atmospheic, and the quantity of gas considered being that which occupies unit of volume at a piessurc of one atmospheie and a temperature of C With these constants the calculated cntical tempeiatiue is 32 C , which agices fairly well wilh the value obseived by Amagat, namely 31 8 C. In other particulars the agreement is less good, thus the calculated critical piessurc is A in] APPLICATIONS TO PARTICULAR FLUIDS 311 61 2 atmosphcies, whereas Amagat's observed value, is 72 9 atmo- spheies. This disci c pa ncy may be due in pait to eiiors of expen- meni arising from the picsence of some air m the gas*, but jt is found that the Van dci Waals equation fails to icprcsent the behaviour of a gas veiy accurately in the neighbourhood of the ciitical point 202. Corresponding States. If we ha\ c two or moie diffeicnt fluids to which Van dcr Waals' equation applies, \\ilh diffeicnt constants foi each fluid, an impoitant relation between them can be established by selecting scales of tempciatuie, picssuie and volume such that the critical tempciatmes of tlic diffeicnt fluids aic expicsseel by the same number, the ciitical piessuics bv the same numbei, and the oiitical volumes by the same numbci Isotheimal curves diawn to these scales foi the diffeicnt (Inids will then coincide in othei woids a single diagiam will show the ic- lation of P to V in all the fluids, when it is icad by icfciencc to Ihc appiopiiatc scales Similaily .1 single diagiam will show the Vmagat cuixes (PV and P) foi all Any point taken in such a diagiam, mtcipietcd on the piopei scale, maiks a definite stale foi each fluid, and foi the dilleienl fluids it maiks what ,11 c called "coiiespondmq stales " The cnlical points for Ihe difltient fluids liuuish m ob\ ions example of coiiespondmg states Thus fluids aie said Lo be at coiiesponduig piessuics when then piesMiies beai the same lalio to the icspcthxc eiitical piessuics lhc\ n ic said to be at eonesponding \oluniesuhen then \olumes beat Ihc same talio lo the icsptcliv e enlical \olumes, and at concspondmg tempeiatuies wlien then lempeiahnes beai the same lalio to the icspectne ciitical lempeiatuus If substances con foi m to a ehaiaeleiistic equation of I he Van dei Waals U p< all Ihicc quantities P, V, and T, simultaneously ha\ e "coiiespondmg" \alucs in the sense hcie defined To put this statement m anothci form, let Ihc unit of. tcmpcialmc chosen foi each fluid be its (absolute) culical tempciatuie, the mill of volume its ciilical volume, and Ihe uml of picssurc ils ciitical piessuic. Then Ihe same family of cmves, eithei on Ihc pi essinc- volume diagiam 01 the Amagat diagiam, will seive to icpicscnt the isothcimals foi all fluids that confoun lo a charactciistic equal ion of the Van cler Waals fvpc Thai this is tiuc of any fluid to which Ihc Van dei Waals * Van dm Waals, Plu/iical Mtimaii s, p 4 OS 312 THERMODYNAMICS [cir equation applies will be seen by i educing' the equation lo a moic general foim Take .any such fluid, in any given state, and wnle its piessure P as p,P e wheie p, is the number by which the picssmc is stated when we use the cntical piessuie P c as the uniL of piessuic. Similaily foi V wnte v,V c -uheie u, is the numbei by which the volume is stated when we use the cntical volume V c as I he unit of volume, and for T wntc t,T c svheie t, is the numbei thai cxpi esses the (absolute) temperatuie when we use the ciitical tempeiatuie T L as unit of temperatuie The quantities p,, z>, and t r arc cnllcd the "i educed" pressme, volume and tempeiatme iespcctivel> . Then T-tT =t - ~ ' c ' 27 bR On substituting these values in Van dei Waals' equation, it will be seen that the constants a, b and 7? cancel onl, and I he equation becomes / \ (/>, + fo) ('', - i) M', 0> 1 - The constants that chaiactciized a paiticulat fluid IM\ <- disap- peaied Accoichnglv this "i educed" chaiactcnstic cqntilion, as il is called is tine of any substance that satisfies a Van dci Waals equation, and consequently the loims ol tlic cmxes roimtcliiin />, , v r and /,__aie the same for all such substances In othei woids, if we compaie any two such substances, using sa\ the tempeiatme and pressiuc as mdepcndcnl vmi.ibles foi the pin pose of specifying the state, and choose "coiiesponfliuo " \ahu-s of the tempeiatme and piessuie foi the two, then the > ohnnes xvill also ha\e "coiiesponding" values This is the theoiem of coriespondmg stales, fust enunciated bv Van dei \VaaIs IL XNas tested by Amagat and found bv him lo be nearly true of a numbei of fluids which he examined lliroui>li a wide lanoe of conditions, and it has been shown lo hold nppioxi- mately in many substances' 1 The validity of Lhe pimciplo docs not depend on the piecise foim of the chaiacleiishc equal ion Van dei Waals' equation is by no means the only one lluiL would lead to the same conclusion. Any charactenstic equation connecting * See S Young, Phil Mag , Fob 1892, also Ins book on Kl APPLICATIONS TO PARTICULAR FLUIDS 313 P, F, and T with no moie than thiee independent constants (two adjustable constants in addition to R), and giving a critical point, can be biought in like mannei to the foim of a "Deduced 1 ' equation in which the constants peculiar to the fluid have chs- Hppeaied Hence any such equation seives equally well as a basis foi the theoiem of coi responding states 203 Van der Waals' Equation only Approximate Useful as Van dci Waals' equation is in exhibiting bioadly the behaviour of a gas even m cxtiemc \anations of state, it cannot be biought by any adjustment of the constants into exact agiecment with the results of cxpcnmcnt It appeals that the actual piopeities of a gas aie too complex to admit of complete statement by the use of s'o small a numbci of constanls. The quantities a and b of the equation aie not stnctly constant, they aie to some extent func- tions of the lempeiatuie, 01 the density, 01 both If constants aie selected which fit obsenations of the compicssibihty, the equation fails to agice \\ilh measuied A allies of the cutical \olume cind cntical picssiue Fmlhci liom Equation (12) ol Ai I 201 uc should expect the latio RT\PV to ha\e at the eiilieal point the value " 01 2 GOT, whatc\ci be the \alucs ol the constants a mid b But the obsenations of Young" 1 show that this lalio is not the same in all gases at the entieal point, lhat in most gases it is about 3 7, but in some it nun be less than 3 5 and in olheis moie than I The iclalions ol the- eiilieal lempcialuic, piessme, and \ olumc <uc m laet less simple than is consistent \\ilh the loimula of A an clei \\.i.ils the ei ideal pomls m dillen nt ae lu.il Minds aie not slnelh "eoi lespondinn" stales, and theie is some dep.iitme 1 1 om I he IhcoK m ol \i I 20'J Vgam, liiUmy Ihc \ r an del Waals equation /' _ _ and dilfeientialmi> \\ilh respect to T, keeping \ r constant, \vc have R ~ V- b which means that when a Van der Waals fluid is heated at constant volume the increment ol piessme per degice of use in tcmpciatme is constant Hence with such a fluid, Avhelhei liquid or gaseous, a thcrmometei of the constant-volume, type would give leadings on the thcrmoelynannc scale without ce>iicction In othei woids the * Lot til 314, THERMODYNAMICS [fii observed piessure coefficient would be independent of the tem- peiatme This ib not true of leal fluids. Again, if a substance confoimed strictly to the equation of Van dei Waals it would follow that K v , the specific heal a I constant volume, would be constant at any one tempciatnie, and would theiefore be the same foi the liquid as foi the gas at any tcmpeia tuie at winch both states of aggiegation aie possible lor taking Eq. (It) and again diffeientiatmg with icspect to T, we have -o Hence by Eq (28), Chap VII, (/) -0 that is lo say K \(.IV / f would be constant at any one tempeiatme This, howevci, is no I confiimed by meassiuements of the specific heat Anothei impoitant paiticular m which Van dci Waals' equation fails to give lesults that agiee with those of experiment is in I he cooling effect of throttling This effect has been measui ed in vai ions gases and 1 ^ apours by expeinnents like the poious-plng cxpcimunls of Joule and Thomson. Such expeuments show that in any real o.is the effect suffeis an inveision when the initial tempciatnie ol llu gas is sufficiently high, that is to say, at high tcmpciatmes the effect of tluotthng is to heat the gas instead of cool it The lad of this inveision can be deduced fiom Hie Van dei Waals equation ' , * To show this we may uv Eq (41 a) of Chap Vff, which o\piessc* tho coolniLf effect in an\ fluid as fdE\ In a Van cler \VaaIs fluid, by Eq (4) above, \ r, ,v \ . Fllrther = ' and by Eq (35), Chap VII Hence we should have dE\ RT a dE _ a dV Adding the two teiras, the whole cooling eflect in a fluid which olio\H Vim b' equation would be BTb By making T suflkiently large the second teim witlun Iho sqiuuo biaokots exceeds the first, wbich means an inversion of the effect When Ihc fluid is a gas at low pressure, and Fis consequently very large compaied with l>, tho oonddion foi inveision is that RTb = 2a in othei words the inversion toinpciaiuio in a gas at very low piessuie would be 2a/Rb vni] APPLICATIONS TO PARTICULAR FLUIDS 315 and to that extent the equation is satisfactory But the amount of the cooling effect in such a gas as cai borne acid, when calculated fiom the Van der Waals equation (with constants which suit the form of the isothermal curves) falls much shoit of the cooling effect that is actually obseived, and if the constants of the equation are adjusted to make the observed and calculated cooling effects agree, then the equation does not accord with the observed figures foi compressibihby* 204 Other Characteristic Equations 1 Clausms, Dieterici Enough has been said to show that Van clei Waals' equation cannot be biought into exact agieement with the deviations fiom Boyle's Law and Joule's Law which arc found in an actual gas. The icason has ah cad y been indicated that the "constants" of the equation aie not stuctly constant. In particular the attraction between the molecules, on which a depends, is piobablv a function of the tcm- peiatme, although it is ticated in the equation as independent of the temperature Various attempts have been made to mod if v the equation so as Lo bung it into closer accoid \\ith the known pio- pcitics of gases None of the^e ha^ e been completely successful in giMng a foimula which will stand all tests lluoughoiit a \ciy wide Kingc of states, though in some icspccls the modified equations approximate bcttci to the obserx ed facts Clausms | gives a characteristic equation which \\e may \vntc in the loini 717- ,/ P - _ (10} 1 ~ v-b Tir + b 1 )* ( h wheic a' and b', as well as b and It, aio constants On com- paring this with Eq (1 o), it will be seen to differ fiom Van cli'i Waals mainly by the presence of T in the dcnommatoi of the last teim, which expresses the addition to P that is due to mter- inoleciilar attractions C'lausms assumes that these altiactions become reduced when the tempeiatine uses, he Iheieby gets an equation which, while it gives to the isolhcimals the same gene rat form as is given by the equation of Van dcr Waals, agrees bctlei with the Joule-Thomson cooling effect When the same method of finding the critical quantities is applied to rt, by wilting dP\ . 7 /&P\ -)= and 7^7-, = 0, dV/ T \dF*J T * See Callendai, Phil Wag , Jan 1903, pp f>S-()0 f Plnl Mag , Juno, 1S80 316 THERMODYNAMICS fen. one finds that V. = 36 -I- 26', T -* /. _ , a'R * ^ 216 (b + &') 3 For caibomc acid Clausius gives his constants the following \ allies R = 003688, b = 000843, a' - 2 0935, 6' -= 000977, Ihe unil of piessure being again one atmospheie, and the quantity of gas considered being that which occupies unit volume at one atmo- sphere and C With these constants the calculated cnticnl tempeiature is 31 C and the calculated ciitical picssmc is 77 atmospheres. Cldusms diaws a theoietical isothennal cmve of piessure and volume for caibomc acid at 13 1 C calculated fiom his foimula This cmve, which is leproduced in fig 9-i, shows the foim assumed by the James Thomson nave in the Clausius type of chaiactenslic equation The hoiuontal stiaight line BC, which exhibits Ihe piocess of liquefaction, is so diawn that the ciest and hollow ol the waxe shall have equal aicas (Ait 199) this considcuilion dilti- nunes its height and therefoie fixes the sanitation j)iessiue The dotted poitions of the cmve exhibit imaqmaiy stales, eompnsed within the chaiacteiistic equation, uhich sen e lo eslablish theoietical continuity between the leal state of homogeneous liquid AH and the leal state of homogeneous vapom CD. A modified and moic geneial type of: Clausius equation is ob- tained bv writing p _ where J (T) is a function of T such as to diminish with perature. In the onginal equation of Clausius, f(T) = . Vnu der Waals has suggested that/(7') may be e ' J \ , wlicie c is 2 7 I S3, the base of the Napienan loganthms, and T t is I lie enlieal lem- peratme. In that case, at the ciitical (empeialun. J (T) would become equal to 1 This foim of characteristic equation A\as aclo[)lcd by Molhc r in calculating his tables of the piopcilics of caibomc ,ieid :| * Molliei, Zcilsrh/ iflf/ii dtc gc^ainmtf Kalla-lndublm, vol n, ISOH and vol ui, 1890 vm] APPLICATIONS TO PARTICULAR FLUIDS 317 -Still another characteristic equation of the same comprehensive kind is that of Dietenci ", who writes P (V - b) = RT^ f . . . (18), \ / \ / * Avhere e is again the number 2*7183, and , b and R are constants. Like the otheis, this formula is founded on the kinetic theory, 80 i i i i i l 70- 60- -j x"" ~"" __^ - o BL / ^"~~^5~ 50 ' 40- 1 / a I / w i / S \ i 30 -Q- \ ' \ / \s 20- 10- Volume i I i I I I 91- TlK.oHlii.al lyoUiciinnl of COj aL 13 1 (.' accoidini; to the equation of f'lausius and like them it reproduces the geneial features of isotheinial curves under all conditions and gjyes a cntical point Since it has only two constants besides R, the pi maple of coiicspondmg states holds good foi the iclation it establishes between P, V and T It makes the critical tempei attire ^O^TT^J ^ ie cntical * Annahn det Physil, vol, v, p 51, 1901 318 THERMODYNAMICS [en volume V c = 2b, and the cutical piessuie P c . Hence at cU the critical point the ratio RT/PV becomes equal to \e- 01 3 695, a value which is in much bettei agreement with obseived icsnlts than \\as the value 2 667 calculated fiom Van dei Waals' equation (Ait 201) In icspect also oi the Joule-Thomson cooling effect and its inversion 1 Dietenci's equation gives a bcttci agieement with expeinnent than does Van dei Waals' A moie geneial form of the Dietenei equation is obl.uncd bj writing T n instead of 2'in the index teim, thus mtioducmg one more adjustable constant - (19) The ciitical tempeiatme then becomes v / y-- . The nimciple of V 4>bR ' l coiiespondmg states uould still apply to any gioup of substances for which a had the same value, since each substance in the gioup would still ha\e only two constants individual to itself. 205 Callendar's Equation. None of these equations is com- pletely successful in lepiesenting the beha\ loui of a lluid in all possible states. But toi the piactical puiposc of enabling tables to be calculated \\hich will show the piopeities of a lluid Ihioiighoiit a limited lange of ranation oj state, it is not impossible to fiamc a chaiactenstic equation which, by empuical adjustment of the con- stants, can be made to apply \\ith sufficient accuiaey and e\cn with gieat accuiaey within that lange, though it m.i\ Kul entnely when caintcl beyond the lange. A conspicuous example ol this less ambitious type of chaiactenstic equation is one which Callenclcii has devised and applied to calculate his tables oi the piopeities of steam | It sen es to expiess \ eiy exacth Lhc obscn eel piopeities of steam within the limits ol piessuie and tenipcKiluie that are usual in steam-engine piactice, but it has no application to highei piessuies, and it makes no attempt to lepicsent Ihe continuity ot the gaseous and liquid states This equation, which Callendai takes as chciiac tens tie, ol tiny vapour, satuiatcd 01 supeiheated, piovidcd the piessuie is much lowei than the cutical piessuie, is RT V- p -~c + b (20), * See Porter, Phil Mag , Apiil 1906 and June 1910 1 Callenclai.P/oc Roy Soc vol 07, p 2CG, 1900, PJnl Hag,Jtui IQQ'd, Bni Articles Theriuodjiiaiuics" and " Vaporisation " vnr] APPLICATIONS TO PAKTICULAR FLUIDS 319 1RT where -^- is, as usual, Lhe ideal volume of a, peii'cct gns b is a constant lepiescnting the co-\olumc s as in olhcr chaiactenstic oqualions, and c is a term which is not constant but is a function of Ibe tcmpciatme Callcndai takes c -, \\hcic C is a constant and ii is a number depending on the nahne of Ihc gas. The teim c repicsents the c flee I of mtei-moleculai loiccs, but instead of ic- gardmg ihcsc loiecs as augmenting the influence of I he cxtcinal piessiuc (which Van dci Waals did by adding the leim to P), Callendai iej)iescn(s by 6 1 Ihen effect in mincing the volume below ils ideal value, in consequence ol the "co-aggiegalion" 01 teni- poiaiy mtcilmknig of some of I he molecules dunng then en- counters Ik calls c the "co-aggiegahon \olume" and tieats it, ai the model, lie piessincs within which he applies Ihc equation, as a I'miclion of the kmpei.ihne only This assumption Mould ue>t be line undei conditions of high elensity, but (01 a gas 01 salinated \,iponr al modeiak piessmes i( gi\cs lesults \\hich ayiee lemaik- abl\ \\<ll \\illi lhos( j ol expciiiutnt lie (oie pioec'iehn^ lo <ip|>l\ r C'alle ndaTs espial ion il mav be iisclnl lo point out its icfaliDii lo lliat of Clausius A\'c max \vnte the cqu ilion ol Clausius (Kq (1<>) of Vit 201) in the foim JfT f^V -^It) 1 ^ P ~ PT(V ,-//p" M; No\\ ,il l<i\\ 01 mod( i.ili piesMiic's Ihc \olume \\ill b< l.uge, and the modilxmg Icinis on tin nyhl \\ill be compaMlM i Ix small \\hen / is laigc MK tlUcl ol the scconel kiiu x\ill not be ninth aikied if r it i ut. 1 lake , , as appioxmiatcly eniuil to ir , and alse> lake P (l r -\-li)- " l V in (hat leim as appiovimakly equal lo JfT/f \\hen Ihesc sub- ihlulions aie made Hit equation becomes JfT _ ' ~ 1> HT*' 1 ' ,vlneh we may wnle in Callendai 's form ., itr c ~~ P ~ 7'" "'" vhcre Ihe more jieneial index it is subslitLiteel I'oi 2 as the index of l\ and C is xvrillen for a' /If Callendai finds that Ihe best agiecment will) obscived lesults, 320 THERMODYNAMICS especially with obseivations of the Joule-Thomson cooling effect, is got by giving to n a value which is not ncccbsauly 2 but may be gieater. 01 less than 2 accoulmg to the natuie of the gas ! . Foi oxygen or mtiogen. 01 hychogen he takes n to be 1 5 foi caibonic acid at low piessine good icsults aic got bv taking it as 2, and foi steam, in the calculation ol his tables, he has taken it as 1 , Itmiibt 1)e emphasized that Callendar's cejuation applies only lo gases and vapouis at low and modciate prcssincs That Ihis is so will be obvious when one consideis the foim of Ihe isothermal lines which it gives on a diagiam of PV and P We may wnlc it Since c is a function of T only, and is there ioie constant along any one isotheimal, this gives 'T Hence m a gas which obeys Callcndafs equal ion the isolhcimal lines would be stiaight, inclined downwaids, wilh incieasing P il cisgicatei than i, and inclined upwaids il b is oiealci Hum c Then. 1 would be no minimum of PV not change ol inclination along an\ isothcimal line The equation theieioie can appl\ onl\ mult i con- ditions such that the hues aie substantially sliaii>hl, nanu'K at low 01 moduatc picssuies htaitmg liom /' -=- llu lints aie in fact neail}. stiaight foi some distance, anil as we saw in \il 1<)7 thev slope down uhcn the tcmpcialme is low and slope up wlu n it is high In an} gas, al a sullu icntly low l( ni]K lahne < isnualti than b, and an isolheimal line theie will slopt do\\n \s the l< m- pcrature inciease,s ioi which the isolheimal isdia\\n,r be conies l< ss, since c = , and a tempciatuu is i cached at which the hue inns level (c = b) Foi am Inghei tcmpciatiue Mian tins I In line s)op< s ii]) like the lines foi hydiogcn in iig 80 The tempeiatuie al which the sign oi the slope changts will he lelatively low in a gas which, like hydiogcn, has a \eiy low eiihcal temperatine, and will be i datively high in a gas like eaibonie acid, as, might be mfeiicd horn the pimciple ot coiic'spondmg stale's In dealing with steam, the limits within which Callcndai has applied his equation aic fiom /eio piessme lo 500 pounds pi r square inch 01 3i atmospheiesf- Within this lange il is not |)iobable thai * Phil Mat/ , Jan 19US, p 95 ( Theciiticalpioasuiein watci-viipoiu IH aliout 20(1 uLiuortplicirH, oi Hay MIX tunes as high as the pleasure up to which Callondai's equation IH lu-UI to apply vm] APPLICATIONS TO PARTICULAR FLUIDS 321 any nnpoitant eiror is mtioducecl by ti eating the isothermals as shaight lines on the diagram of PV and P. Besides lepiesenting accuiateh/ the behaviour of the substance within this lange, when suitable values aie chosen for the con- stanls, Callendars equation has the very convenient pioperty that diffcicnlial e\picssions deduced i'lom it foi thcvaiious quantities E, I, 0, K ]l} K v and so foith, by applying the gencial theimo- dynannc relations of Chap VII, lake f'oims such as may be icadily intcgiated Ilcncc it enables niimencal values of these quantities to be calculated, to any desncd numbci of figuics, which will be theimodvnamically consistent with one another II would lie possible to fix the constants by icfcience only to 2xpuimcnts on the compicssibihly of the gas at vaiious tempera- tines, if sulfieicnlly accmate data of that kind weic available But Callendai picfeis (o (i\ Ihem by lefcu-nce mamlvto obseivcd values )i the Joule-Thomson cooling effect Then iclalion to the cooling effect will be appaient fiom what follows 206 Deductions from the Callendar Equation. Taking Hit Jdllcndai equation, mcc c , vc have dc nC nc . (Vc n (i , and - = - dT 7 1 "' 1 7" dT* c 1 dc d ic\ Vlso, since , } i - , rr dT ( T ) = ' )ilfciciiliating the equation with icspect to T, keeping P constant, dy\ _ /.' _ dc If nc \dfJv~ P dT ~ P "'" T hfleicntialing- with icspect to P, keeping T constant, low by Eq. (27) of Chap VII, in any fluid ~dPj^ ~ I (dT^Jp 2L 322 THERMODYNAMICS . [cu. Hence in a gas 01 vapoui to which Callendai's equation applies. (dK,\ _n(n+^)c ( dP ) T ~ T ( ^> Integrating, we have n (u+1) cP A p = -y + A ( 2( >), wheie KJ is the constant of mtegiation It is the limiting value of the specific heat K }} when P = 0, at the Icmpcialme T lint since any gas in that infinitely laiefied condition may be healed as peifect, Callendar assumes that K J} ' may be taken as having the same \alue at all tcmpciatmcs to which the equation is applied It should be noticed that this mUyiadon is peifouned along an ibotheimal line, and that the const, ml ol mlegialion is not necebsaiily the same ibi othci tempci.ilmes To deal K ,' as con- stant when the tempeiatuie is vaned Iheieloit invokes ,m .issnmp- tion \vhich is independent of anvllung m llu equation itself Again, by Eq (4-1) of Chap VII -\\c had loi Hit mtasint ol (he Joule-Thomson cooling effect in am fluid dT\ Hence for a gas to which the Callcntlai ttpial ion applies, llu eoohn effect is RT - p V nc - / = V H- c - b -! uc - = ( + !)< -b As was explained m Ail 182, () is (he toll ol Umpeialuu pei unit fall of piessure whc>n the gas passes thionnh ., , )()1011S |)Iu , 01 any other thiottlmo device, and p is Lhc qmmlily of lual II J Avould sene to maiaUim Lhc ono, n<1 { (oinpcialuic, il it wei. supplied from outside dining the pioccss Eiom the above lesul it follows that Callendni's foinmla piovirtcs foi Ihe nueision of th, cooling effect -which is known to occur m leal gases When (// -|- 1 ) is greatei than b the expiession loi- p ,s posilivc, (he s is thei cooled by thiottlmg. This is the usual case But when (n + 1) c j Jess than b, p , s negative, the gas is then waimcd by throlllmo u vin I APPLICATIONS TO PARTICULAR FLUIDS 323 hydiogen is at ordinary tempeiatures, and as any gas will be if the initial tempciatmc is sufficient^ high. By laismg the initial tem- C peiature the quantity (11 -f 1) c is leducecl, since c = Inveision of the Joule-Thomson effect occurs when (n + 1) c = b, or when nc = c + b But, as we saw above, c + b is the slope of any fd(PV}\ ' isothcimal line on the diagram of PV and P, namely I -775 I \ (LI j 'i< Ilcncc if Ihe isolheimal slopes up \\ith a giadient steeper than nc Lhe Joule-Thomson died uill be a healing, if il slopes up less steeply than Lhis, 01 runs level, 01 slopes down, Ihe ellecl will be a cooling It will be apparent fiom llie'sc consideialions lhat mcasinc- ments of the cooling dfeel linnish an impoitanl means of sell ling the \ allies of the constants in ('allendaTs equation, apai t lioni dnec I dc leinu- nation of Ihe isolliuinal IIIRS Cal- lend.u in fael .issnmes Ihal Iheco- \olumc b is equal lo Ihe volume which \\e>uld be occupied il Ihe gas weie all condensed to a liquid, and (hen calculates the values ol n and c I mm observations ol Ihe cooling died ' An ilhishalion ma\. help lo make some ol Ihe above pomls clear In lig !)5, which is a diagiam ol Ihe Amagal t\ pc, \\ilh PT and P lor co- oieim.ilcs isolhemials au skdche'd (not lo scale) for a <>as obeying Calkndar \ ecmalion Thev, aie, as we saw, stiaighl lines within Ihe langc to which the equation is applied AS is an isofherm.il drawn foi a tempciatme such that Lhc.va.pom becomes satuialcd at a moderate pressure, which is issumcd to be uilhm. the langc of pressure for which Lhe equation Iroldsgood Aceoidingly Ihelrnc J^issbiarght, up to Lhe saLura Lion poinL 6'. The ciuvcd line through S is a poition of the' boundary nrive, below which lies the "wet" region, where the beginning ol w w Pressure Aiim^ni isollic i n \r Id ('llllc'lldlirH U'l IHlH * Phil May , Jan 1903, p 87. 324 THERMODYNAMICS ft-ir condensation would be repiesented bj' a \ r eitical straighl lint 1 , #ir, P as well as T being then constant AS slopes dowmv.uds, and the effect of throtthng,at that temper atuic, is lo cool I lie gas I'fi' is an- other isothermal, drawn for a lo\\cr tcmpoiuLiue, lo which I he same remarks apply. The effect of throttling rs si ill lo cool I he gas al I he higher temperature for which the hon/onlal isolhennal />/> is di.nvn ( c = &) and at any temperature up to th.nl of ('(', which is I he isothermal corresponding lo the imeision ol UK Joulc-Thoinsoii effect, namely that for which (n + 1) c = b, tlie npwaid gi.idie'iil of CC being equal to nc At any highci lenipeiahiie, such .is thai for which DD is drawn, the upwaid giadicnl is slerpei and I he effect of throttling is to heal the gas 207. The Specrfic Heats m Callendar's Equation The t \- piession given in Eq (26) foi K !t , in a gas that conloinis lo Callendar's equation, enables the specific heal <il const. ml pitssiiu to be calculated for any temperature and am piessuu willnn I lu range to which the equation applies, when llu \aluc ol l\ lt ' (,is- snmed constant) foi the qiven gas is kno\\n as \vc'll ,is UK < onslaiils of the chaiactenstic equation In oielei lo ohl.nn ,i conespoiKlinn expression for the t.peei fie heal atconsl.inl \ohini( il is most con- venient to write the characteristic equation in UK lonn PU - RT, where U stands for V b + c U is a Junction ol / ,md 7' only Differentiating with respect to T, keeping V eonslanl, rr ( dP \ T, (<IU\ U -,- + P( R \dTJr \tlTJi I-'M AT- 1, ' -\dT), \,ir But (g) _*_ e . , K , ,<';") .'" "("I. 1 ''' \al/ r rfjf / \dJ-l, (IT- /'- Substrtutrug these values, andrcmembeiing lhal ' = , ,, we obl.un from Eq (28), y M U'Jr'7'l 1 - 1 - (/) ( 2 ). Then from Eq (29), Pnc f ^ %nc\ __ Rm / (I vnr] APPLICATIONS TO PARTICULAR FLUIDS 325 Now by Eq (23) oPChap VII, m any fluid. dV T Hence in a gas 01 vapour Lo which Calendar's equation applies, < 30 > In Integra ting we have to icmembci llutal conslanl tempeiaLure dU = dV. Accoidingly, Rue / nc\ A = I // 1 -- ) -|- constant. Willing tins in the i'onn *.- -'(- I-) HA".' (81). we sec that I he conslanl ol inlegiation K u ' is the limiting \alne of K,, when 7 J - 0, \\lnch (like K ' Ail 200) is iakcn as having the same \alue il all U mpualincs lo ulnch Ihc equation is applied NL\I, to (ind an evpussion Joi K K, ]i\ Eq (2S) ol Chap. He nee in a gas lo which CallcndaTs equalioii applies we obtain the lelalion , ,., I (52), by subslihiling I he \ahn.s aheidv found in l^<]s (2S a) and (22) Poi Ihcsc l\\o dil(< Kiilial eoc Ilieii nls In the lunil \\hen I' =- I he \olinnc 1 beeonies indcliiiilely gicat, Ihc kun . \ anishes, and K; - K V ' ~ jt (3.$), as we should c\pccl Itotn the fact I hat I his gas is I hen lobe regauled as peileel (eonipaie Ait ISO) II should be nolul (hat Ihc assump- tion lhal A'/ is conslanl letjuues I ha. I K ' should also be conslant 208 The Entropy, Energy, and Total Heat, m Callendar's Equation To find an expiession lor the entropy we shall apply Eq (20) of Chap VII which is line of any fluid, 320 THERMODYNAMICS [en Here, and in what iollows, the icicloi A, which is 1 /J (Ail 1 1), is mtioduced in older that heat quantities (including R) may be uumeiically stated in thermal units. By Eqs. (26) and (22), Ait 206, we have, using thcimal units, An (n + 1) cP , , L (dV\ _ R Anc K = T + * \dTj P ~ P + "T~ in any gas that obeys Calendar's equation Hence in any such gas AY An (n + 1) cP , R dp _ Anc d6 = -~ dl -I --- ^2 -- IL ~ p T~ ' (31) Integrating, this gives 'LncP </> = K v ' log, T - R log, P - - + 7? (35). or </> = 7v ; log, T - R loo c P - t H - 2? (35 ), wheie 7J is the constant of inlegiation. To find an expiession foi the inteinal eneigy E we may mosl conveniently use the geneial equation (S) of Chap VII, dE = Ttty - IPcJV In a gas that satisfies Callendai's equation By substituting this and the value of <7(/> in Eq (31) -\\o h.n e dE = (K ; - R) dT -|- An ( nc dT - cdP\ \ ~*~ J = K v 'dT -And(cP) (3(,), fiom which E - A',/ T - AinP + B' (37), wheie R' is the constant of inlcgialion NoU I ha I Lhr inUiual eneigy falls shoit of the value il would have in a peifccl gas by the amount AncP. To find the total heat we have, by definition / -- E -\- APF Hence, fiom Eqs (37) and (20 a), m a Callendai gas / - (K v r + R) T - A (n -1- 1) cP + AbP -h B' = K P 'T - A [(n + l)c-b]P + B' (38) Fuithei, since A [(n -f 1) c b] is, by Eq (27), equal to the Joule- vnr] APPLICATIONS TO PARTICULAR FLUIDS 327 Thomson cooling effccL p cxpicsscd in theimal units per unit chop oi picssure, we may wnlc this expression 1'or the total heat in the foim I = K,'T-pP+B' (38 a) On diffeientiatmg Eqs. (37) and (38) with respect to P, keeping T constant, we have Anc and ~ A These icsults agicc wilh the cxpicssions ahcady gi\cn iov the cooling effect The whole cooling effect is - (-. = } . by Ail 182 \dP' T J it is made np of Fin thci, since f J is constant foi any one tcmpeiatuie, the slope of any constant-tcmperatuic line is constant., on a chait of / and P foi steam (compaic AiL 102) The hues slope downwaids, with ineicasmg P, and llic slope is less at high tcmpcialnics, since < is then less To complete Ihe list, expulsions mav be added foi the Junction (\\lucli is G) and Hie hmction i/r, in a Callendai gas These aic found at once horn Ihe above lesults = A'/ T (1 - l<,n ( 7') -i- RT IOO L p-j( c - I,) p __ HT -I /?' ifi = E- T(f> - K,,' T - K J T lon i T \ UT 1< )i> t /' - JIT -i //' ( 10) All the foieoomo <h duel ions liom C'.illc nd.n's cf|iialion hold good foi any gas 01 \apoin to which the C'(|ii<itiou applies, \vhalc\ei b( Ihe \alucs ot the constants, piovidecl the specific heal at /eio i)ies- sme may be taken as independent, ot the tempei.il me within Ihe langc ol apphe<ilion 209 Application to Steam In applying Ins equation to steam, Callcndai assigns to the constant n a value such that nil = K v ' This iclalion, which is not true for all gases, gives a value ol n Joi steam thai is consistent with the ob.se ived el'lccts of tluotlhng It has the practical advantage of allowing expressions for the bc- haviom of the gas dining adiabahc changes to take a veiy simple form 328 THERMODYNAMICS [en. When K v ' = nR it follows fiom Eq (33) that Ay = (n + 1) R, and in that case the expiession for cj>, Eq. (35 a), becomes = ( n + 1) R log c T - R loo t P - P (41). Now in achabatic expansion lemains constant, and LhaL can p happen in this expiession only if - hl is constant Hence in the adiabatic expansion of a Callcndai gas in which the i elation nR = K v ' holds, the piessuie and tempciatuie aie conncclcd b} the equation p yw. = constanl (42) p (Y _ M Fuither, it follows that in all such cases - - is constant dming adiabatic expansion, because by the charactcustic cqualion we have P (V - b) _ P T ~ T >1 " l ' and under the condition stated both leims on the ii<.'ht-h<ind side aie constant. Again, undei the same condition that nR = K v ', T n *P(V - b) -p- - T = conslanl, I whence T (V - b}" = conslanl ( l.J), i u i i P(V-b) and, multiplj ing b> -- - , n j-1 P(V - b) " = constant (IJ) All these Jesuits for adiabatic expansion aie line of steam, willnn the limited lange thiouoh which Callendai s cqualion is ai>])hcal)lc They hold good so long as the substance icmains in the homo- geneous state of a gas, uhdhei supeiheatcd, salnrated, 01 super- cooled (Ait 79), and they cease to apply when pai t oi it liquefies In the calculation of his steam tables Callondar lakes foi Ihe numeiical value of the co-volume b the volume ol unit mass of water at C , namely 01G02 cubic feet per Ib. Foi R he takes 11012 in mean caloues, corresponding to 1-982 per niol, and equivalent to 151 17 foot-pounds pei Ib Foi n he takes \ This vm] APPLICATIONS TO PARTICULAR FLUIDS 329 fig me is based mainly on thiottJmg experiments by Ginidley **, Peakc[, and Callcndcir himself | lie lakes fov C a value such that c is 4213 cubic loot al 100 C. This makes 157 52 x 10 C=Q 1.213(373 1) '', or 15752x10, and c = - |in Ilcncc when V is the \ olumc of 1 Ib. in cubic feet, P is the picssuic in pounds pei squaic foot, and T is the absolute tcmpciatme in ccntigiade dcgices, the Callendar equation j>y r = - c -i- b becomes, for diy slcam in any state, 1 T I 1 7 7' 1 T7 *V> v 1 F = - f -I-001C02 (L5). p r * As a numciical example, let it be lequned to find the \olumc of 1 Ib of steam at a ])iessiue of 100 pounds pci squaie inch and a L-mpeiatmcof 2 10 C. lieu P = 100 x 111 and T = 513 1, making }' = ] ;J733 - () I ISO -|- Oll0 - 1 2J37 cub It. This will be found to agice \villi the value m Table C ( Yppcndix III) \\hcic lhe\ oh i me is labuLiled toi vanous picssuus and for tempcia- tmcs langmg fioni 100 C down lo llic Icmpeialmc of saturation and belou it The \ oliinu's below llic kinpciahueol satmation icfei to \\ilei-\ajK)iii in a supc icooltd (melastablc) sttile, such as that which is sol up by adiabatic e\jiansion in the absence ol nuclei on which condensation niav occm. In this example the slcam is shghllv siipc t he alcd, llic salmalion lemjieialuie loi a piessuie of KM) pounds bcmn pisl uiidei 230 C Callcndai also labiiLtli s sepaialt l\ I he "co-Jg>icy<ition \oliime" c loi \aiious Umpi'ialmes Some ol I he values aie yivcn In. low volume c /( Dn/ Steam in tint/ ttffilc, in cubic led [)c\ Ib Pomj) i 1Vni|> ( Tom IP i Toin[) f I 102 70 5570 1 10 1000 210 1780 10 1 () r >7 SO o 5o<>i 150 02771 220 o i <)<:* 20 00117 00 K.I! 1()0 25<>2 230 1555 30 H4JO 100 1213 170 2375 210 1 150 10 7557 1 10 o 3sr>7 180 220J 250 OlSdG 50 OSO 1 1JO WO 190 2010 2(50 12S2 00 0147 130 3255 200 1009 270 1205 * Phil Tintit A, vnl 10 1, p 1,1000 | Pine Tioif Sot A, vol 7fi, p 185, l ( )0fi | ScoBunkwotUi, Phil Tunis A, vol 215,]) 383,15)10 330 THERMODYNAMICS [en la furthci illustiation of Eq. (J5) vanous isothcinial lines Jtoi steam aie diawn to scale m fig. 96, showing PV in lelahon Lo P as 500C 800 700 __400"C 600 : 30gc- 500 400 300 200 100 Pressure, Pounds per Sy Inch 100 200 300 ~400 500 Fig 9(3 Isotheinuils for steam, fiom Callciul.n'H eyiuiLioii calculated from that equation Here the pressuies aie expiosscd in pounds per sq inch, consequently the nuniencdl values oi PV m] APPLICATIONS TO PARTICULAR FLUIDS 331 jven 111 the figure must be multiplied by I'll li it is dcsned to ha\ i i hem m foot-pounds The dolled continuations of the isotheimal nes ioi 200 and 100 below the satuiation cmve lepiesent values 31 supei cooled 01, as it is sometimes called, supcisatmated vapour 'he full lines diawn at constant piessurc lepresent the first stages i the condensation of a wet mixtuie It will be obseucd that at he highest temperatuie at which Callcndai applies his foimula a steam, namely 500 C , the isolheimal still slopes down with ici easing P Thioughout the whole woilcmg range the thiottlmg f steam produces a cooling effect. Since the value assigned to R it> 11012 caloiy, and that of n , \- 3 the i elation K v ' = nR icqunes that K e ' bliall be 036707 nd A"/, which is K v ' + If, shall be 17719 We have next to show how the tabulated values of the total heat ic calculated The foimula for /, Eq (38), becomes, Ioi steam in ny homogeneous condition, whctliei supci healed, saluialcd, 01 upeicooled, j ^ R ^ T _ { ( i^ _ b} p ,. Jr ivuio., in caloncs pei II) , / = () 1771!) T _ C'ArJi) 7 ' .| u> (|0), hue c and b ate expiesscd m cubic feet pci Ib ind P in j)oiinds ci sq loot To obtain a numeiical value Ioi /?', which comes in as a conslaiit f mki>iation, we musl fix some /uo slalc fioiu \\hiih Ihc lolal ( at ol the substance is lo be icckoiiLcI, 01, whal comes lo Hie same img, we must assign a uuuKiieal \alnc lo llu lolal heat m SOUK nown slate In llic calculation ol lus l.ibks Callt ndai assume s that le tolal heal of \\atei is /eio at C and is 100 al 1()0 U C 1 , nuclei ituialion pressuie m each case ' * r l'lm insiniipluHi nnt onl\ li\is llic /(uo fidin \\lmli lliu Ldliil liml IN tu IK c'koncd, bill alyn i;i\os Lo tin 1 thci nial mill n \aluo \ciy slililly ^noaloi lluui Ilio iMii onloiy ,;^ flc'liiad in Ail 13 The tluiin.il anil ol ( '.illi luku'w lahlis and 'inuiliiH is one liuiirlicdlli of I hi 1 Llnin^c in lnl.il heal \\\\u li w.ili-i iiii(lci()c-< \\lii n n licak'd 1 1 (Jin () Q (n I OK 1 midoi Uici (rai yniu) ii< ssmo ol nalu ration, \\lic KMH I ho duiaiv moan ca-loiy is OUP ImiulKHllli of Ilio change' in liil.il Jic^at \\liin \\.ilci i^ iatod lluoui,'li I hi 1 aiinio inLeival of (.oin|)0i,iimo nndci a loiiHtunt [inwsinc ol 0110 .mosphcro Calk nclai's unit" 11 tin. laigei of the two by about one part in fom lousiuul Thin difleinnce i ol no pi actual nnpoitancc il IH so Hinall as to bo \\ill rtluu tho limits oi CHOI ot cvpeiirnont The fifruios Ioi Ilio mc'( hanical t'limvak-nl In'al gi\c-n jn Ar t I i i plate, slu< tly, to Ilio laigoi unit, whu b is tho one iiHc'd in to tablos Tallondai takes his caloiy to be equivalent lo -4 ISdK / 1U 7 CI^H, tho 11 1 csponding value ot the constant piosmne moan caloijr would bo J 18H8 x 10 7 oi{j;s Tho relation between the two iimtM will Ijo intido i loai if we \vnlo out an onoig} 332 THERMODYNAMICS [en In passing fiom the stcite of water at 100 C In thai ol di\ saturated steam at 100, undei a constant pressuic oJ' one atmo- sphere, the fluid takes in 539 30 calones, that being the latent heat as determined by experiment. Hence / foi steam A\hen T is 373 1 and P is 14 6S9 x 144 is 639 30 The value of c at thai Icmpcialme is 4213, and b is 01602. Substituting these figures in the ex- pression for /, Eq. (46), we have 639 3 - 1TT19 X 373 1 - . 14 + fl , fiom which B' = 463 995. Foi most purposes B' is taken as 464 The foimula foi the total heat of si earn, m any condition within the Callendar lange, accoidmgly becomes - 719 T - - -I- 4.C-1 H) Values of / calculated in this way aie given in the tables As an example, take the same case as befoie, namely steam al a picssme of 400 pounds, per sq inch and a tempeiatuie of 240 C With these data the numbeis aie / = 241 84 - 25 20 + 464 = OS3 55 The tabulated value (Table D) is 6S3 54 account for the piocess of wanning water Let i 7 ,, be the internal e IK ILT\ of unit mass of water at C and the conespondmg saturation picssme P () , whidi is H02 / UJ pounds per sq ft , and let / be the total heat in that slate /=/,' I J /' \ Lei E IM be the internal eneigy of watei at 100 f and the coiicspondmg s.itninhon piessuie P 100 , which is one atmosphoie 01 14 GS9 , 111 pounds pei sq It , and h I 7 ino be its total heat ui that state I im =E 1M +Al\ (M ] Inn I n is (lie \olnnii al, 0C , namely 016 cub ft , and V 1IU is the \olume at 100 Imamo tlio wnlci, initially at and P to be uncl^i a piston Incicaso the load on tlu pislon lill tin piessure is P )00 Since watei is almost inconipicsMblc this cloc-H not Hensibly cluing the volume, 01 the tempeiatuie, 01 the intunal cnng}, wludi may be (akin an still equal to E (l Then heat the w, ate i nuclei constant picssiuo to 100" (Ins it-qmiLS the addition of 100 constant pressuic caloiies In being heated the watei uvpaiidn from V to F 100 and thciefoie does woik on the jjiston equal to AP l{m ( V im - F ) Hence the net gain of inteinal eneig^ in the whole opeial ion, (<\i>irss( fl in connlant- piessure calones, is r _ 1ftl }) nr from which or 7 ino - / = 1 00 + 1 ( P 100 - P ) F,, - 1 00 023 Thus the same change of total heat winch is measuicd bv 100 ol C'allc-ndai's units is measured by 100 023 constant-pleasure units in] APPLICATIONS TO PARTICULAR FLUIDS 333 It follows from Eq. (37) that in steam 10 cP = 36709T - + 164 (47). 'he above expicssions are in tciins of P and T. We may also vpiess both E and I foi steam in teims of P and V, eliminating T mcc K ' = nit, Eq. (37) may be written E = nRT - AncP + Ii' nt by the c'hriractenslic equation (20 a), wJicn R is expiessed m icimal units, RT AcP = AP (V b) Hence, in steam, E = AnP(V-b) + 13' (18), Inch gives, in thennal units, Uciin /, being equal to E -+ APV , becomes / -= J(n + 1)/'F- AnbP \- B' (J9), ving, m thcinial units, i \pv - o w -(-JO I- (19). 100 V ; This leluLion nitiv be wntlcn in the foim 00128 (W) :ncc if we use /> to denote the piessiue in pounds pci squaie incli, e volume, in cubic feet, of I Ib ol steam in any diy state, supei- atcd, satuiated, 01 snpeicooled, is given by the formula v _ 2_2i.36 (1 - 404) + This affords a convenient means of calculating the volume when c total heat is known. Take dgam the same example as befoic am at 100 pounds per squaie inch and 240 C The tabulated hie of / is 683 54. Substituting it m the formula we find V to be J314 + 0123 = ] 2137 cubic ieet, in agreement with the figure t from Eq (45) and with the tabula led value of V 334 THERMODYNAMICS [en. By differentiating Eq (49) with icspect to P, keeping V constant, we obtain, in steam, 1T\ = A(n+l)V-Aub (51), which is constant foi any chosen value of F It follows that lines of constant volume on a steam chait with P and / foi cooidmalcs aie stiaight in the legion of supcihcal ', as in fig. 33 (Ait 102). We shall next obtain a Moikmg loimula foi the cnhopv ol steam in any diy state, b} using Eq (35) and finding (he value ol the constant B The constant is found bv woikmg ou(, horn indepen- dent data, the entiopy oi sal mated steam at 100 C Following the usual convention the enliop\ ol icnfci at C is taken as zcio It follows fiom uhat is knoun about the specific heat of watei, as will beshoivn in thcne\l ailicle, that the enliopy of watei at 100 C and a picssuic ol one atmosphere, is 0. Til, So In passing at that constant piessmc liom the slate ol walci <il 100 to the state of satin a ted steam, the substance takes in 5.19 .5 units of heat at the absolute tcmpeiatuic 373 1 its cnliopv the le loir 539 3 mcieascs to 31 ISO + -- oi 1 75732 At that lempciatuie c is 373 1 4213, and P is 1 1 GS9 x 1 H Hence, by Kq (3 j), 1 75732 = 17719 log, 373 1-0 11012 log, (1 1 689 x 1 1 I ) _ 10 [2]3 - " '' ()SO "I" 3~ 373 1 y 1 100 ~ ' Flora which B = _ 021901 Substituting this, and inlie)diicing the f.uloi 2 3025S5 to comeil common to Napierian logaiithms, the Jonuula Joi the cnliopyol dry steam m any state becomes = 1 09876 Iog 10 T - 2535(5 log 1(J P - 0023S1 ' - 21901! (52) Values of d> aic given in Ihc tables (Table K) fei the same range of piessme and tempeiaturc as was used m tabulating T and J. 210 Total Heat and Entropy of Water It is known from the icsearches of Regnault and others Ihat Ihe specific heat of water is not constant, but increases with using temperature. Callcndai * In the wet region tho constant- volume hnoa leinam voiy nciuly shaight, for tho above relation still holds with legaid to that pait of tho steam which is uncon- densecl, and its volume constitutes neaily the whole volume of tho wot mrctme in] APPLICATIONS TO PARTICULAR FLUIDS 335 uggests* that this inciease may be due to the piesence of water- apour dissoh r ed in the watci. He supposes that when watei and s vapoui aie in eqiulibiiuni at any tempeiatinc a volume of the a pour equal to the volume of the watei is contained within the r atcr Consequently when water is heated its total heat mcicascs lore lapidly than it Avoulcl do if the specific heat weie constant, >i the heal that is icqmrcd lo I'oim the dissolved vapour becomes laigei piopoition of Lhc whole heat This thcoiy gues icsults r hich aic consistent with Ilie experimental d.ita, and Callcndar dopls il in calculating, loi his lublcs, Ihc total heat and the nhopy of walcr It has the aehantagc of allowing each of these wo quantities to be expicssed in a simple manuei Lcl J\ be the volume of 1 lb of sahnatcd steam al anv assigned 'mpciatuie T, and let V m be the volume ot 1 lb of walei at ic same lempeiatuie and picssuie Then accoidmg lo Ihe u'oiv, 1 II) of *' walei" in that stale is leal!}' 1 lb of a solution, )nlaimni> chssohcd vapom , Ihe walci conceals willim it a olumc of sahualcd steam equal lo /' ', If the icmamdei \\eie Ko tinned into vapoui, undei coiiilanl piessiuc, we should June tolal volume of vapoui equal to V m + (J\ - J r w ) 01 V\. and ic heat taken in dining Ihe piocess would be (he latent heat L LV , [(nee "' icpicsents the heat that is icquued to pioducc Ihe ' S ~ * II' ipoui iiiihallv pies(iil m the walci bcloie Ihe 1 loiuiriiion ot any pai.ili st(am begins This heal had lo be supplied \\lulc the- alei was berny wanned up lo Ihc tempeialuu ol satiualion, il insliluU^ a pail of (he lolal heal ol walei /, Tin olhei (and elue-l) pail of (he. lolal heal ol w t itei i 1 - supposcel > HICK asc at a iiiiilonn lale is Ihe le mpeial me i ises it may llicie- ne be leptc'smlcd by h. (T T (] ), where it is a conslanl anel 7' is Ihc excess of lempcialmc ,ibo\c (' , \\luch is lakui as 10 stailing-|)oint in icckoniii" Ihe te>tal heat Hence, adding Ihc vo ])aits, Ihe tolal heat of watei undci satuialion piessme al ly LempeiaLine T is r 7-r TV ~~1"- __ "'" T \ \ - 1 OJ "I" TT ir ir T y ' ' ' cie L n , F H)O and F s i cfei to the state at C. At tha t tempeiatuic ic latent heat is 59 I- 27, the volume of water is 016, and that of Luraled steam is 3726 cub ft Hence v ~~_- -= 0029 calory SD io * Phil Tians A, vol 190, p 147,1902 336 THERMODYNAMICS [en. This is the constant which has to be subtracted to make I u = when the temperatme is C. To calculate K, we have /, = 100 when T - T = 100, by Ail 209. L is then 539 3, V w is 01071 and F u is 26 789 Hence _M9 Sx 01071 _ from which K = 9966(5. The foimula Foi /, tlius becomes I w = 99666 (T - 273 1) + J^jf ~ <> <><>-} (53 a) " > 'u> Values of /, calculated by this foimula aic given in I he tables. V Thioughout the woikmg lange the ratio _'" is vciy neailv ' S ' !(, the same as V W \V ^ and ils numci ical value is appio\imalclv equal to 00004p, wheic p is the sahualion piLssiiu in pounds per sq. inch To find the entiopy of watei undei salutation picssmt <il <mv tempeiatiue T, we may think of the walci as bung biounhl lo ils actual condition by two slciye.s Imayine it to be hc.ilrd lf> lluit tempeiatuie in an '"ideal" mannei, naniclv \\illiouL I he loim ilion of anj^ dibsolved steam, and then the dissohed sic am lo IK lonned at that temperatuic The entiopy depends only on UK ,11 lu.il con- dition (Ait 14) Taking, as bcfoie, the cntiop\ ot \\alu at C' to be zeio, we theicfoie have = 99666 log, ~ + _, /5- Jl V , ~ 00001 (5 I ), * (' i, ~ ' ial which is the foimula used by Callendai foi the enliopy ol watci undei satiuation picssme' 1 It follows that the value ol G, 01 TV/> /, loi walei undei sahna- tion pi assure is G w = K T log f ^ - OOOOlT - K (T - T () ) -\ 003 (55), * o the term v l " cancelling out " S " 10 211. Relation of Pressure to Temperature in Saturated Steam. A formula connecting the piessurc with the lempeiature of steam in the satiuated state is rnosL easily obtained by making * Sluim Talks, p 7. [i I APPLICATIONS TO PARTICULAR FLUIDS j of the fact tJial G, 01 T(j> - I, has the same Milne toi the urated \apom, at any tcmpcialme, as loi the liquid at the same npciatwc and picssiue (Ails 90 and 185) Ey Eq (39) of Art 8 the \alne ol G foi steam in any state is Ay T loo. T ~ Ay T - HT Ioy t P + A (c - b} P + BT - B' nee for diy steam at saturation pi cosine 7 J S - A'/ T loo. T - A'/ T - RT loo. 1\ + A (c - b) 1\ + BT - B' (50) ice G b - G to \\e oel, by equaling (55) with (5(1), t i . li\ p ~ - B -I- /c - Ay -I- K loy e T + 00001 oi> - This c \piesMon allows the saLuiaLion piessuie P., to be iound lor / lcni])t i aline On t>mn<.> the \anons eonstanls the values eacly slaletl, il becomes 101 J l(n i P - V ' " = 5 S!JO!)I - " ' - - 51') 17 loo. T lendai ' pills llns m a loim moie smLibk loi e.ilenl.ilion, b\ (sliditmy 2 o()'2,")S'j lo^,,, loi lo^ L , and 1 I ly;, loi l\, />, biin^ I lie malion pussiiiL m pounds pel squaie inch 105? (< - h] /> 2)()'J .5') i/'s~- \,, - J107II')- - I 717.51 li)i) |(| 2' (57 h) 1 s.ilmalion pussine eoiiesponduin lo any oi\tn Umpti time ioiind b\ woikniLj out the iinhl-hand side ol llie eqiulion 1 then .ul|iislinn MK \aluc t)t loy p^ unlil llu l\\o sides bceome lal. 'he picssiiu s ol satmaled sltam, ihns cl educed J'lom (.'.ilIendaTs lac'lcii-ilic ecpuilion, ayiee \ r eiy closely, Ihioii^houl Ihe lanyc ivhich I lie equation applies, with those mcasmed by Rennault, I Ihe anicement beh\cen the ealeulaLed and measnied iii>mes is lenee ol Ihe soundness ol CallendaiAs melhod Finlhei con- lalion is oblamed when the \olumes, as calculated by Ins alion, aie conip.ucd with experimentally niuasuicd volumes h of sal mated and ol snpcrhcutcd steam * MLUIH Tahiti,, p 27 E T 22 338 THERMODYNAMICS . [en 212 Formulas for the Latent Heat of Steam, and for the Volume of a Wet Mixture Fiom the equations L = I, - I a and / = Kt + ~ "p- - 003, ' s ' w where / is the tempeiature on the centigiatlc scale, we lui\ e or L = (I, - ><T + 003) (l - p"J (58), which Callendar p wntes V\ 1 chopping the 003 as negligible in this calculation For the \olume of a wet mixtuic, V v (Ail 71), hi gives the foimula ' V I Kt F / -// (r>0) To obtain this ^ e ha\ c M'w *i ' //' on again dioppmg the 003, also 1, = L + I al = L + id -l- ' "' ' s ' ii' I f . Kt qL (V ^ T lr } H- />/ Hence _ f = J ~<~\,'~ ~\^ ~\ I / r r 213 Collected Formulas for Steam For c'on\ UIH-IK-C ol icfcicnce and use tlic foimnlas aic oolleolc-d IICMO h\ r nuansol \\Inch the quantities in the Steam Tables may be cah-ulaU d In these foimulas V is the volume ol 1 II) in cubic Jcct, /* is Ihc picssme in pounds pci squaic foot, and p in pound 1 , pci squaic inch Centigiade degrees aic used in the reckoning ol Icnipcialnrc and quantities of heat T is the absolute lenipeiature and / the tcmperatuie fiom C. The following expicssions foi V, /, E and r/> apply lo chy steam * ftlram Tablet,, p 10 air] APPLICATIONS TO PARTICULAR FLUIDS 339 n any state, that is to say, supciheated, saturated, 01 supeicooled, bul not to a mixtuic of steam and Avatcr The volume 15 , ,,., ,-- r> UG Vss l*Ll.J _lo/o-xlO ^, 2 ,';- ^ ^ The lot.il heat ,, , 40 1 . (10 a). .vheic c is Llie ' co-agyicyaliou volume" in cubic feet, namely 15753 XO" ,373 ' T ' V T EquaLion conned my the volume \viLh the picssuie and total heat- r = ** l V--) + 00123 (06). y> The internal encigy \()P(l' - 010) ,,, , '"- .. n M01 < 1So > I lie cnlio|)v cP /, 1 0<>S?<>l<>n l() r-0 J,5.J50l()o 1(J / J -0 002381 -0 21964- (52) The following c({ii.ilion, which applies only when the sleani is il salnialion [ju'ssine, deltimincs the icLihon ol picssiuc lo lempualme in s.ilui.ited slcam >( )03 .V) - '21 071 l ( > - " - I 71731 lo"!,,'/ 1 ( r vr/;). \Vlitn tlie satmalioii piesMiie lot an\ i>i\en tcmpcialuic T has been delrimincd l)\ nn.uis ol ^lns einialion. Hie \olunic, lolal lieat, _ > ncii>\ ind enliopv of salm.iU'd slcain at that lenipciatuu (l'\, I , J<^ and c/> 4 ) aic, lound b\ Ihe abn\c loinuil.is The lalcnt heal L =(!,- f )00(>0/ -I- 003) (l - '" ) (58), \ ' s ' \vlu b ie V m is the Aolunie of 1 Ib of walci at saluialion ])iessuie. \Vithin the ranye usual in piactice, Ihe lalio V w \V a is veiy ncaily .qual to 0000 ly; s , wheie p, is the salutation piessure in pounds pei square inch, and the working ibimula is L'-=(I o -() 990CG/) (1-0 0000 l-p a ) 340 THERMODYNAMICS |c The total heat of watei at satuiation piessme /, = 99G66/ + j^^jr - 003 ' X t, ' in 01, veiy ncaily, withm the working langc, I IL -= 99666* + 0000 lp,L The entropy of \\atei at saturation picssruc The function G, uhich is Trj> - I Foi chy steam in any state, G = K ] ;Tloo t T-(K ] ;-B)r-l!T\o^P + AP(c-b)-Jr (39), which gives G = 1 09S77T Iog 10 T - 696832' - 25350?' loi> 1( , P + *<'-'>">_ l()l I 100 Foi satuiatecl steam, 01 \\aLcr at saLuialion pic-ssuic, 01 a niixlino of uatcr and steam in eqiuhbinmi, T G b = G, e = K T Ioy t - - : (T - T n ) - 000017' | 003 (55), * o which gn cs G, = G w = 2 29I9T lou 1(J ^ _ o 9)0(i(,/ - ( ^"'J' 214 Tables of the Properties of Steam Tin Slr.im T.il>ks in the Appendix contain some lepicscnlalivo nunihc-is, l)iil iclci- ence should be made to Callendai's Tables I'oi a nioic cout])hlt scl Tables A and B relate to the special case ol MtLiualtd sli'ani A\'hc n steam is satuiated a single piopeily, such as oithc'i I IK- Icnipcialuic 01 the piessine, fixes ils state In Table A I he j>iop(.ily which is assumed to be kno\\n is the tempeiatmc, and Ihe labk' yi\ cs coi ic>- spondmg values of thepiessiuc, volume, lotal hc.il, and cnliopy all foi the satuiated state It also gives the latent heal Snmlail} Table A' gives the volume, total heat and enliopy of ualei at saturation piessuie It also gi\ cs the funcliou G, which is Ihe same for water and for saturated steam. In Table 13 Ihe piopei ly which is assumed to be known is the piessine, and coiiesponding values aie given of theothei piopeities in the saturated state, namely the temperature, volume, total heat, entropy, latent heat, and the function G vmj APPLICATIONS TO PARTICULAR FLUIDS 341 Tables C 5 D and E vela I. e to the geneial case ol steam in any dry 5 Late, whether superheated or snpei cooled A knowledge of two propcitics is then icquncd to specify the state, the two that aie Delected as independent variables in the tables aie the tempciature ind the picssmc. Table C gives Lhe volume, Table D the total icat, and Table E the cntiopy, in i elation to these two variables [n each table a, zig-zag line indicates (he bonndaiy between Lhe aipcihcatcd state (abcAe) and the supcicooled state (belou). In 'tossing this line the substance passes thiongh the state of satura- 1011 Fiom Table D it is easy to find Ihe heal of foimalion, nuclei onstant piessuie, ol' steam m any condition of supciheal The otal heat, at tlie gneu picssme and tcmpcialiue, is obtained fiom he table, and Ihe heat of IbunaLion is found bv subtiactnig liom hat Ihc total heat of watci, al Ihe same piessuie and at Ihe empeiatuie ol I he leed Again, liom Table- 1), \alues mav be infcncd of the specific heat K ;( ) ol steam <il U mpe lalmes and picssiucs \\itluu Ihe uingc of he table- K tl leu <in\ condition of the steam is equal to the amount }\ Mhich / mcicasLs pel degie'e ol use 1 111 Icmpc lal me, at constant )iessuic The change ol / pe'i ek'giee is louud bv noting, m the ppiopuatc piessuie column, Ibe 1 amomil b\ \\hich / change's foi u mlei\al ol 10'\ and ch\ idmg lluit b\ 10, this ^i\i.s the mean alue of K tl o\ e i the mUtxal, which is pi act icallv the same Hung s K lt at the middle 1 lempe laluie N'aluc-^ of llu specific heal al anous constant |)it ssiues .mdloi \ <u ions lempc laliues li.n c been educed m (his \va\ r liom I lie tabulaled x.ilui.'s ol Ihe total heal, nd aie gi\en scpai.ileh m Table I ( ' 'IMie /ig-/ag hue IKIS Ihe same leaning as in Ihe olhci tables, llu- ligmcs abo\e it i< l.ilc lo siipci- caled sleam They sho\\ r a dcci ease ol A^xvilh using Icmpcialiuc, ut at luglu'i lempe latuies (bcvond I he lange lo which Calleiulai's q nation applies) Iheie is a maiked incicasc, as \vas pomleel out i Chap VI APPENDIX 1 EFFECTS OF SURFACE TENSION ON CONDENSATION AND EBULLITION 215. Nature of Surface Tension In Ails 135-138 it was pointed out that when watei -\apouv is suddenly expanded iL assumes a mctastable state, becoming supci swindled owing lo wh.it was there called a static ictnidation m the I'm niation of diops Wilson's experiments weie cited to show that, in Hie absence ol' foieion nuclei, a vapom will become much supcisaluialed l>cToic chops will form, and it M r as mentioned thai I his is an cITecl ol sin lace tension in the liquid In tins note some account will be gi\cu of what is meant by sin face tension, and how il ulaids [he loiinahon of drops in a cooled vapoui , also how it letaids Ihc loiinalion of bubbles within a liquid when the liquid is boiled The cohesive foices which Ihe molecules of any liquid c\crl upon one anothei make the fice sin face of Ihe liquid Inline as if il weie a stretched clastic skin. Il is to I his lhal Ihc phenomena of capillanly are due the use of a liquid column in a lube \\hen Ihe liquid is one that wels il, and the dcpicssion of Ihc column \\hen the liquid does not wet the tube To this also aie due Ihe hums assumed by liquid films and by chops II is Ihc lension of Hie surface layei that makes a chop lake a sphcncal shape when Iheic aie no chstnibing foiccs the chops of mollcu melal in a shol- tower, for example, become sphcies as they fall ficcly, and solidify into sphcncal shot befoie they icach the bollom A chop of mei 1 - cuiy on a plate, 01 of clew on a leaf winch il does nol wcl, would be sphencal were it not foi the upwaid picssuic of the suppoil on which the chop icsls, the smallci Hie chop is Ihe neaici docs it come to being a spheie, foi the distmbing foice due lo the weight is relatively ummpoitant in a small chop As a resull of smface tension, the cneigy contained in a chop of liquid is giealei lhan the eneigy contained in an equal quantity when lhal loims pait of a big mass of the same liquid at the same tempcialuic, foi energy is stoied in the surface layer in much the same A\ay as it would be stoied by the sti etching of an elastic skin u'P.i] EFFECTS OF SURFACE TENSION 3i3 We are concerned heie only with theimodynamic aspects of sur- dcc tension, and especially with its influence on the formation of hops in an expanding vapour. We shall see that, as a consequence if suiface tension, a small chop will evaporate into an atmospheie if snpci saturated \apoui, because the vapour piessure which is cquncd to picvcnL cvapoiation fiom the cui\ed suiface of a diop s gicatci than the vapoui prcssuie which is sufficient to pi event vapoiation fiom a flat suiface of the same liquid at the same cmpeiature, in otlici Avoids, that at any given tempeialure the aluuition piessuic foi a small diop is greater than the noimal atmation pressure. The film that is fonned when a soap-bubble is blown, 01 when a oapy liquid is smeaicd over a ung or hoop of wire, consists of two ui face laycis, back to back, with some of the liquid between. Vhcn the film is vciy thin, as, foi instance, when it looks black in cficctcd light just befoic it breaks, it may be said to consist of \vo suiface laycis onh , but it can be made a hundred 01 moie imcs thicket than that and ha\e just the same tension, foi the tate of tension exists in the sin face la>eis only The tension of uch a film, \\liethei thick 01 thin, is the tension of h\o suiface iveis, in othci wouls, if is t\\ice the suiface tension The tension 11 a liquid him chlfeis fiom fhat of a slictchcd sheet of mdia-iubbei i ollu'i clastic 1 mcmbinnc 1 m these impoilant icspccts it does not lunge \\hen the him c'outiacls 01 is si i etched, and il has nccessanly lie same \aluc in all diicetions along llic suiface Let a liquid film he I mined on a U-shapcel fi-une (fig 97) by kctling a wiie .111 with I he 1 iqmd, placing it o\ c\ C\ and hen diawing il away in fhc , ^ luccLion of I lie anow The oicc Ihal will ha\c lo be ppheel Lo duiw it away 01 to Q \ told it fiom coming back is !iS7 whcic I is the length Ali Liiel S is the tension of the in face layer on each side of " he film pci unit of length Fi" 97 Hie quantity S so defined s iH'asuics the surface tension of the liquid In chawing the loel iway tlnough any distance x m the dnection of the ariow the voik done is 26*^ Hence S also measuies the woik clone in 344 THERMODYNAMICS [AFP forming a single suiface layer, per unit of aica of the layer; m othei woids, S measiues the potential cneigy thai is sloud m each unit of aiea of the free suiface of a liquid in con-sequence of its surface tension. It follows that the siufacc eneigy of a sphencal diop (that Is lo say the potential cneigy vuuch is clue to its siufacc tension) it, 4>7T)*S wheie ? is the laclius oi the diop. The spherical foim winch a fiee diop assumes is Ihc Joim winch will make the suiface enemy (foi a given volume) ,i minimum. A chop resting on a suppoit takes such a foim as will make ils lolal potential eneigy a minimum, namely Ihc sum of Lhc cncigy of suiface tension and the eneigy of position which the chop has in consequence of the height of its centie oi i>ia\ily alnnc (he level ot the suppoit 216 Need of a Nucleus Imagine a diopio beo apoialmi> nuclei conditions that keep its tempeiatuie constant Encit>\ lias lc> be supplied in piopoition to its loss of mass Lo pio\idc foi I he Lit cut heat of the vapoui that is foimed But the chop is losinn sui ffjce eneigy in consequence of its diminution ol suiface, and to sonic extent this reduction of suiface cnciyy supplies Ihc I.itenl Iic.it that is lequiied, only the lemamdei has Lo be supplied hoin oul- side the chop Consequently a chop is moie u-ad\ Lo (\aporalc than the same liquid in bulk, at the same tcnipciaLuie, and it will continue to e\apoiatemtoan atmosphcic \\lucli \\ould be sal 111*1 1 rd with respect to the same liquid in bulk Moicoxu, .is the 1 diop gets smaller and smallci (if we assume lhat the leduclion ol' si/c ma\ go on without altennq the natuie of smfaee (tiision), a sln^c Mould be icached when the loss of potential cneiuy due lo con- tiaction of the suiface would become sufficient to supply ,ill I lie latent heat of the vapour that is passing ofl In lh.it c \ciil, no heat Mould have to be supplied fiom outside Lo complete Lhc c\apoiation of the diop it would become mheieiiLly unstable and A\oulcl flash into \apour Foi the same icason a chop cannot foim excc-pl aiound a, miolc'us, and the laigei the nucleus the moie icadily iL loims When paiticles of dust aie piesent in expanding vapoui, the liisl cliops to be foimed use them loi nuclei, as was shown by Aitkcn (Art. 79), and only a small amount of supeisatuialion occurs be! ore such diops begin to foim The cloud of paiticles obscived by Wilson when dust-fiee air containing Avatei -vapour is expanded enough to i] EFFECTS OF SURFACE TENSION 345 cause an eight-fold supersatmation aie foimed aiound much s mallei nuclei which consist, piobably, of accidental co-aggiega- tions of the molecules of the gas itself, 01 of electrically charged molecules, such as aic always piesent in small numbeis It should be added lhat the presence of an electnc charge gieatly favouis condensation of the vapoui upon any nucleus As an electufied diop cvapoiatcs, the chaige lemains behind, the potential eneigy due toclcctniication theiefoie mcieases as the drop becomes smaller, foi I he cncigy due to a constant chaige vanes mveiselv as the ladius of I he sphcie that carncs it In this icspect the effect of an electnc chaige is opposite to Lhat of surface tension. Hence when a chop is chaigcd moic eneigy has to be supplied Iiom outside to make it cvapoiatc than Mould beicqunccl it it weie unchaiged An electn- cally chaigcd drop will theiefoie cvapoiate less icadily than an unchaiged diop of the same size, and maygiow laigei in an atmo- sphcie that is but little supei saturated 01 e\ en not supeisatuiated at all In \apoiu \\luch is slightly supeisatuiated it is found that an^ "loni/mg" action, such as that ot an electnc spaik, 01 of Uonlgcn lays 01 ot ultia\ lulet In-lit, \\hich sets fiee the ions 01 pai ticks coineyuit" unbound electnc chaigcs, bungs about a cloud ot condeusalion, by ci eating fiesh nuclei, or )n stimulating the pouus ol existing nuclei thiongh causing them to acqimc an eketiic chaige ' 217 Kelvin's Principle Confining oui attention, howc\ei, to diojjs which aic not cleclncally chaiged, \\eshall no\\ considei how, as a conseqiK IKC ol smlaee tension, the eqiulibiuim of a chop ot gi\iu si/e clc])tncls on I he slate oi supeisatiuation of the \apoui aionncl it Assume I hi liquid and the vapoui to be at the same [emperatmc Liquid with a Hat suiface is in cquihbiium with the \apom abo\e it when the vapoui is at the picssiuc. of saturation Ihc ir is lliui no tendency on the \\holc foi the liquid to e\apoiate OL loi the \apoiu to condense, any evapoiation that occuis being eviclJy balanced by an equal amount of condensation Liquid in Ihe lonn ol a small chop is, owing to its cmvccl suriace, in equi- hbnum with the suiioundmg vapoui only when the piessuie ot the \apoia suiiounding- it exceeds the nomial satmation piessiue by a (k'hnile amount, in othci woids, only when the vapour is supei- saluraUd The degree of supeisatiuation iieccssaiy foi cqui- libiium depends, on the curvature of the smface, and theiefoie on * Htei Ni ,1 J r riiomson, On the Condiution of Ekdncity though Oases, Cliap VI I, C T R WilTOii, Phil TUM* A.,\ol l'J2, 1839 346 THERMODYNAMICS | MM- the smallness of the drop This principle Avas fust established by Loid Kelvin ! . It is of fundamental impoitance in explaining ihe retaided condensation of expanding steam We may apply Kelvin's gencial method as follows to find a iclation between P s the normal piessuic of satin ation (which is Ihc equilibrium vapom-piessuie ovei the flat sin face of a liquid | ) and P' the eqmhbimm vapoin-piessiuc OA ei a cmvcd MM lace, such as- the suiface of a i>mall diop. Take foi this put pose Ihc tuivcd suiface at A, fig. 9S, \\hich is foimcd by holding in the liquid a capillaiy tube of a """) matenal such that the liquid does not we I ^ it The column of liquid m the tube is accoichnglydepiessed thiough some distance h, and if the boie is small enough the ficc ft suiface at 4 is sensibly pait of a spheic -_= -^^-^=- Imagme the liquid to be contained in a \ closed vessel, and that the space C above I it contains nothing but the Aapom of the I liquid Let all be at one tempeiatuic T. h The "tthole system is in equilibiium 0\ci ' the fiat suiface at B theie is Aapom \\liost 1 ' pressuie is P s ovei the cun ed suilaec at , A theie is \ r apoui of a highci piessuic P' . ^_ The difference P' P s is equal to the weight of the column of \apom in the lube (pei unit aiea of cioss section) fiom the level of A to the level of B Let a be the weight of unit volume of the A apoui 11 ,, it ^ this weie constant, the \\ eight of Ihe column of \apom in the tube (per unit aiea of section) would be simply crh But a depends upon the piessnie P, it is equal lo I//" and may theiefoie be wntten p a = KT> if we take the equation PV= RT to apply. The chifcicncc in Ihe t^o vapoui piessuies is p> _ p _ r^^ mtegiated between the level at B and the level at A Compaie next the hydiostatic piessuies within the liquid just h PIOL Roy So( Echn ^ol \ar, 1870, Populm lect'ne* andAddn sac*, vo] 1,11 d I \ Namelj', the satiu f ifcioa pressure tor anj assigned tcmpoiatuio as given in tables of Ihe piopeilies of satiuated steam il EFFECTS OF SURFACE TENSION 317 undei the surface at B and iusL nuclei the smface at A. Just under the flat sin face at B the hydiostatic pressure in the liquid is equal to the piessure of the vapour ovei the suiface; il is theiefoie equal to P s . As we go down Lhiough the liquid to the level of A, the hydiostatic picssiuc inci eases by the amount ph, where p i^ the weight of unit volume of the liquid Theiefoie just undei the cuived suiface at A its value is P^ + ph. But we may also calculate the hydrostatic piessuie under the cuived suiface at A in anothei way The Lop of the liquid column at A, which has a suiface layei m tension, may be tieated as a segment of a spheie of laclius ;. lib surface la\cr foims a cap whose suiface tension S causes it to pi ess down upon I he liquid belo\\ with a pressmc p such that m-p = 277/5". That this is so will be seen at once bv considmng the cquihbiium ol a complete hcnnspheie of the t>amc cunalurc and with the same suiface tension. Round the cuciimfeicnce (2n)) of the hoiizoutal plane foiming the l)asc of such a hennspheic time \vould be a \eitical foice 27T/.S' balancing the icsiiltaut foice due to the piessuie p acting on the aica of the base, 7r> 2 . Hence 2,9 P- , , and the hydiostatic piessuie pi^t undei Lhe cm\ r ed suiface is theiefoie equal to 9$ P' + / Equating Ihc h\o cxpiessions for this h\ dm^lalic piessuie, we have 1 >,V P' + j - J\+pli, 01 ' 2<S = p h-(P' - 1\}. ) r Hence, since P' P s = $adh, 2S = ph $adh ~ \(p a) (III. And since dP = adh, 2S ( p 'p - a ._ [ p 'p |D . - = - - dP = - dP veiy neaily, ? 'r* 'PR " because a is small compared with p. On substituting P/RT for a this appi oximation gives 2S l p 'dP P' * 6 P' \fT _ c P 348 THERMODYNAMICS [VPP This applies to any liquid suifacc whose radius of cui\ al lire is i It therefoic expiesses the iclation of the piessiue P' m tlic supcr- saturated vapour aiound aspheiical diop of ladius > to Lhc noimal pressine of saturation P s (over a flat sniface) Toi the same Lempci a- ture, when the diop is in equilibrium, in the sense that" iL is ncithci giowing by condensation nor shimking by evapoialion Tlic expression shows ho\\ the dcoue of supeisatuiation P'H\ ncccssai y foi the equilibiium of the diop mcieascs when the size of the diop is i educed For a diop of gi\ en ladnis anv mcicase of P abo\ e the value so calculated would cause Lhc diop to giou The expression also shows what is the least size oi diop thai can exist in an atmospheie with a gn en dcgiec of supcisatui.nl ion an> diop foi which i is smaller A\ould chsappeai by c\apoiation Ib is only when the diop is very small that the excess of /" ovei P s is at all considciable This is best sho\\n by numtiical examples If \\e take watei-vapoui at 10 C 01 '28-3 absolute, and usec g s units, RT (which is ticaled as equal to PT) is 1 30 H)' 1 The sui face tension of watei at lhatlcmpcialuie is about 70 dynes pei hneai centimetie,and/-jis 1 qiammcpci cubiccentimclic Hence , P' 2s 76 I 01 [(")(-) == 010 P, 130 )- 10 s / 2303' D ' where D is the chametei of the diop in milhonths of a millimctte The foimula accoidmgh gives these icsults Ratio of Vtipoiu pussiiii 1 Diameter uf diop in equilibiium \vitli the in inilhonllis of clif>i) to nunn.il -,alui.i <i nulbmetic tmn pio-,siiic fni iho^aiiu ttinpi i.itino (P /Pj 100 1 02 r )0 1 O r > 10 1 2b r ) 1 f,9 2 .32 1 102 This means, for instance, that a diop of watci two milhonths of a millimetie in diameter vill giow if the latio of supcisatuialion in the vapour around it is gioatci than 3 2, buL will evapoialc iJ that latio is less. Hence when the idtio of siipcrsatmalion is 3 2, drops uill not foim unless theie aie nuclei picscnl which aic al least big enough to be equivalent to sphcics with a diameter of two milhonths of a, millimetie. In Wilson's experiments a cloud of mist was produced when the supeisatuiation was S, which coiicsponds, by the foimula, to a * To comett from Napienan to common ij EFFECTS OF SURFACE TENS JON 349 diameter of about 1 1 milhonths of amilhmetie On the assumption that the foimula may still be applied to such small nuclei, it might be mfeirecl, if theie were no loiuzatjon, that water-vapour contains many nuclei of that oidci of magnitude, which may be pans or small gioups of molecules co-aggregated by chance encounteis. IL will be obvious fiom Kelvin's punciplc that a chop of water cannot continue to exist in an atmosphcie of sal mated vapour When the diop and the almosphei'e aie at the same temperature, the diop can exist only if the atmospheic aiound it is super- saturated. Foi any given degiec of siipeisatuiation theie is a value of / (determined by the foimula) such that a chop of smaller radius will evaporate and a drop of laigei lachus will glow Thus the bigger drops in a cloud Mill tend to giow at the expense of the smaller chops. 218 Ebullition Similar considerations go\ern the foi matron of bubbles in a boiling liquid We may tieat any small bubble as a sphencal space of lachus >, containing gas, bounded by a spherical envelope in which theie is surlace tension Outside of that is the liquid, at a pressure P In consequence ol the surface tension in the envelope, the pressure inside Ihc bubble P t must exceed P bv. the amount 2S/), \vhcre A' is the surface tension in Ihe boundarj surface of Ihc bubble, making <?V I* _ p - 1 / \Vhen / is very small this implies a gieat excess ol pressure within the bubble If no pai tides of air or othei nuclei vuic present to start the formation ol bubbles, boiling \vould not begin until the temperature \scie raised much above the point coi responding to the external picssuie, and would occui \\itli almost explosive violence Once formed a bubble would be highlv unstable, for as the radius increases the tension ol' the envelope becomes less and lest, able to balance Ihc excess of pressure within it This happens, to some extent, \\hcn \\ater is boiled after being ficed of an in solution it is then said to boil with bumping It follows that a pure liquid may be superheated, that is to say, ] a i seel above Ihc temperature of saturation corresponding to the actual pressure. Tins is an example of a metastablc state like the state that is produced when a vapour is supercooled without condensing, or when a liquid is supercooled wrthout solidifying Water at atmospheric pressure may be heated to 180 C or nrore when it has been freed of air and when it is kept from contact with 350 THERMODYNAMICS [AIT i the sides of the vessel by supporting it m oil oi its own dcusiLy, so that the Avatcr takes Lhe ibim of a large globule immciscd in oil In the ordinal y piocess of boiling, a bubble contains in general some air 01 other gas besides the vapour of the liquid iLsell. With- out gas in it, the bubble could not exist in stable equilibimm With gas in it, the bubble will be m stable eqmlibiiuin when the paitial pressiue due to the gas piOMdes the necessary excess of the whole internal piessure P i ovei the external piessine P Any i eduction of the bubble's size would then raise Lhc piessine of I he gas more than enough to balance the increase of 26'/j. Let P v be the vapoiu -pressure inside the bubble If we assume that Lhe external pressure and temperature remain constant, the pailial piebsme due to the gas ma}' be expressed as a/> 3 whcie a is a con- stant. Then P l = P v -I <7/> 3 , and the equation deteimmes the \alue of / at which the bubble is in equilibrium The quantity P v P is the excess of the A apoui-picssiuc in the bubble o\er the pressure in the liquid Dillcientiatmg this with respect to ;, to find the limiting condition loi stabihh , \\e ha\ e , A\ lien , - , i i 3 3 / and therefore when P u - P = ol Hence foi stability P V P must be less than 4>S/'3t. Tins means that \\hcn a liquid containing bubbles of ladius > it. heated, the temperature will use until the \apour-piessurc within the bubbles exceeds the pressure in the liquid by the amount t , but when that point is leached Lhe bubbles will become unstable and ebullition, will begin Callendai ' calculates on this b.isis IhaL Avith bubbles one millimetre in diamclci water (under one atmospheie) will boil at a tempciature of 100 05 C , and that to pioduce 10 of supeiheaL the diameter 1 of the bubbles must not be more than about - ^ mm. * Enc Bid , AiLiclc 'Yapouzation " APPENDIX It MOLECULAR THEORY OF GASES 219 Pressure due to Molecular Impacts Accoi cling to the moleciilui thcoiy, a gas consists of a \ ei y large numbei of paiticles called molecules men ing with great velocity Each molecule moves freely, with umfoim velocity m a shaight line, except when it cncounleis anolhci molecule 01 the wall of the containing vessel In an cncountci the velocity changes in chiection, and gcncially in amount, but theie is no dissipation of encigy, the mole- cules bcha\ e like pcifecth elasLic bodies As a icsult of manv cncountcis, a stable chstiibution of speed among the mole- cuks is established but the speed of any one molecule is being cotislaullv changed, b^ its cncountcis, within vuy \\idc limits. The length ol llu 1'ieo path, \\huh il ha\ciscs between one en- couulci ,ind Hie nc\l, is also quite nicgulai The a\ciagc of that leniilh, 01 Mhal is called Ihe ' mean fice palh," is \ en long coni- paud \\ith the dune usions of the molecule, ilsc'lf This chaiacLeiislic distinguishes a i>as horn a lujuid Jn a, gas the a\ e.iagc Lime dining uluch a molecule is moving in ils lice path is \ei\ laige compaicd \\ith Mie dme of an cncoimlci ]3\ r the lime oJ an encounter is mcanl Ihi lime dining which the molecule is cithci in conlact wilh anothci, 01 so ncai it that thcie is a sensible (nice acting be [\\een them "When a gas is coinpu'ssed, Die mean liee path is icduccd, .u id the eneounteis become nioic lietjuctit bct\\cen one molecule and anolhei and also be I ween Ihe molecules and [he 1 \\alls ol the \csscl When a gas is healed the speed with which Ihe molecules mo\e is incieased, we shall see, immediately thai then a\ eiage kinclie enci<>^ is piopoilioiul to the leinpciatmc The molcculai thcoiy is now well established theie is conclusive evidence that actual gases do consist ot pailiclcs moMiig in the mannci which the Ihcoiy piescubcs The ]}icssmc of the gas, Ihat is lo say, the picssure which the gas excits on every unit of smface of the containing vessel, is clue cntiiely to the blows of the molecules upon the surface the mo- mentum given to the sinface by their blows, per unit of area and pei unit of time, measuies the pressure m kinetic units. 352 THERMODYNAMICS [M>P In any gas thai is chemically homogeneous, all I he molecules have the same mass Call that mass m. Lei N be the numbei of molecules piesent m unit of \olume ol' tho gas in any actual sLutc as to piessme and tcmpeiatuie Then inN icpiesents Lhe densiLy, namely, the whole mass per unit ot volume, and V, Die \oluuio pci unit of mass, is equal to 1/inN Befoie piocccding to considei the picssinc caused bj molecular blows, we shall make the following postulates, (1) That the molecules aic pcil'cctly lice except dining cn- counteis, and thciefoic move m stiaight lines with uinloim A elocitj , fiom one encounter to the next, (2) That the time dining which an cncountei Lists is negligibly small in compaiison with the time dining which the molecule is fiee, (3) That the dimensions of a molecule aie negligibly small in compaiison Auth the fiee path These tlnee postulates, aic equivalent lo assuming that Lhe gas is pei feet in the sense of Ait 18 Tile's aic not shicllv ti uc of any ical gas, but we shall assume them to be line m what immediately iollows, and shall theiebj deduce Uom the molcculai theoiy a lesult which coiiesponds to the ideal loimula l*V ^ RT Suppose the gas to be in ccmilibuum m a vessel at usl, anel lei Ihe velocity v ol any molecule be lesolxcel mlo icclangiil.n com- ponents v^, v u and v,, along tluec fixe el <i\cs Considei the piessme due to molecuLu blows upon a containing wall, of aiea S, fanning a plane suiface al nglil angles to the dnection of x The conliibution which any molecule makes to the piessme on that wall is due cntncly to Ihe component \elocily v, nothing is contnbuted by the components v a 01 v. Any molecule which stnkcs the wall has the noimal component ol Us velocils icvciscd bj the collision lie nee the momenlum due to Ihe blow is 2i>iVj_ \\hcic i\ is Ihe noimal component of I lie \eloeily anel in is the mass of the molecule. Considei next how to expiess the sum ol the e fleets ol such blows in a given time Foi tins puiposc we may think ol the mole- cules as divided into gioups aceoieling to Ihcn \elocihes at any instant. Let n be the number, m unit volume ol the gas, whose ^-component of velocity, v^, has the same numeiical value Since the numbei of molecules is very gicat, we may take the munbci to be the same m one cubic inch (say) as in anothci. Thcic will of comse be very many such groups, each with a diffeient \aluc of c] MOLECULAR THEORY OF GASES 353 t Think, in the (list place, only of those in the gioup n. Half of he whole numbei of molecules in the gioup aie moving towaids ', the other half arc moving away fiom it At any instant of time lieic Avill thcicfoie be within a small distance Sa' of the smface S, nd moving towaicls iL with component Aclocity i\, a inimbei of lolccules of that gioup equal to }n$Sd\ A molecule distant S,r om >, and having a component velocity v^ towaids >S', would ^\ idch S in a time St --- J piovided it did not encounter any othci lolcculc on its way Ilence the number of blows dehveied to S y molecules of that gioup, in the time Sf, would (on the same LO\ISO) be equal to the numbei of such molecules a,s onginally y within a distance S,t, namely the number \nS8x Hence also I he momentum due to the blows on the aica S in the me 8t would be equal to \nS5iU ,c SiflVj , which becomes, pei unit aiea and pci unit of tune, & nitn\ ~. -- nmi\-, of BK v ^Wt This is UK momentum contiibutul by one yioup only The essiue P is nude up of Ihe MUH of llic quaulitics of momentum nlnbiited l)^ all the yioups, hence UK N is ,is btlou llu \\holc numbei of molecules pi.i unit of liime, and i 1 ,- is Ihc 1 t i\ i I<IL>C of v,~ loi all Ihe molceulcs Now I lie \eloe ily i' of an\ molecule is icLited to its components the equation ..> _ .. j , i , , 2 I - ( , I" l'^ I L y i nce, it u r c \\tiley-foi the a\ ciagc value of i "loi all the molecules, v 2=s w, 3 + v y -+ v, 2 = 3v, 2 , cc the motions take place equally in all diicclions The square loot of v- is called the " velocity of mean squaic " II lot the same thing as the average velocity, but is the velocity a 'lecule would have whose kinetic eneigy is equal to the average ictic energy of all the molecules The expression foi P may therefore be wiitten P = 354 THERMODYNAMICS |APP. FmHiei, since mN is the quantity of gas in unit" volume, 01 1/F, where V is (as usual) the volume of unit mass, this gives PV = \v*. In obtaining this result we made (in oidei to simplify I lie aigu- menl) a proviso that each molecule of a pnilic-iilai gioup, lying initially within the distance ,r of tlic wall, shuck I he wall without encountering othci molecules on the way. This is not line, hut any encouulci on the Avay docs not alfcct Lhc final lesnll in a gas to uhich the thiee postulates apply. For in any cneountei, some momentum, peipcndicnLu 1 to the wall, is simply tiausfc'i ted to anothei molecule, and i caches the wall \\ilhoul loss The molecule which takes it up has to tiavcl the full icmamdei of the distance in the clucction of .r, neithci moie noi less, since I he dimensions of the molecules aic negligibly small (Poslulale ,'*), and no him is lost in the encountci (Postulate 2) Hi nee Hie geneial usull ot the cnconntcis is not to altci the amount <>l monunlum \\lnch leaches the wall in any given time, and the conclusion u mains valid that PV =- 'w 3 C'ompaimg this \\iih Lhe pcifccl-yas equation PV= RT we t,ec that u 2 is piopoitional to I he absoluk U mpc lalmc. , and consequently the a\ciagc kinetic cncigy \\lneh UK mohciilcs possess m Mituc of then \clocily of hanslahon is pmpoi honal lo the absolute tcmpciatmc We shall call then entity ol haiislalion E' , they may, in addition, ha\e cncigy of olhu kinds The encigy of translation of the molecule s 7i" is ei|iial lo \c~ pel unit mass of the gas. Hence by the moleeuku lluoiy PV _ 2 7," L V iJ'j , and the picssinc is equal Lo two-thuds ol the cm my of tiauslalion, pei unit volume of the gas. It may be noted in passing thai I he moleeulai llieoiy explains why a gas is heated by compicssion Think of I he gas .is contained in a cjlmdci, and being compicsscd by the pushing in ol a piston Then any molecule which stukcs Ihe piston recoils with an increased velocity because it has struck a body that is ad\ ancing towards it. The component \elocity v y normal to the piston is not simply leversed by the blow, but is increased by an amount 2z/, where v r is the velocity with which the piston is moving when the molecule c] MOLECULAR THEORY OF GASES 355 tnkcs it, for the quantity which is ic^evsccl is the iclative cloc'ity v^ + v'. The icsult is that the motion of the piston in omprcssmg the gas augments the a\ eiage velocity of the molecules, nd consequently mci cases v' 2 , on which the temperature depends. 220 Boyle's, Avogadro's, and Dalton's Laws These laws )llow from the molecnUi Iheorv, foi gases I hat obey the thice Obliilates Keeping v 2 constant, we have the law ol Bo} r lc, 'V = constant, since PV }v* If Iheic ,ue two gases at the same piessnic, since 7* ]//>A 7 y 2 1 CMchl "'i^Vi 2 ' - M 2 N 2 v 2 * a\\u'Il has sliown that if h\o gases aie at the same Lem- lalme, the average kinetic cneigy of a molecule is the same both, 01 -7 , ninY =- w 2 zv encc li they aie al Hie saint piessuie and the saint Lcmpcraliue A\ = A',, <il is lo sav, Ihe nuinbei ol inoleeuk s in unit \olinnc is the same i bolh, \\lnch is A\oi>adm\ Law IL lollems lhat the density, 01 rissol unit \ ohiinr 1 , difk i s in I he t \\o Base's in the laliooi I lie masses then molecules, 01, in olhu \\oids, the elensit\ is piopoilional Hie inoIeenLii \\eiyhl (Vit 15S) : Again, the nioli enl.n theoi\ sho\\s lhal in a nnvtiiie of t\\o 01 >ie gases, e.teli ol uliich obe\ s (he thie'< k postulates, /' =-- \niiN f\* -I .'/ ,A r ^ -I- etc ol In t \\oids, I lu ]iai lidl piessuie due to each eonslil ne'iit of Ihc \luie is the sanu .is il nonld be il the othei conslitucnls weic t theie This is in agieeiuent \\ilh Dalton's Law (Art (}'2). Tin.' unmix i j\ r ol niolmilis pci culm if nliiwf 10, v\liic-li H Ihc s.iino Fot Moiilr fiii^fH n(. llu Sdiiif lnii|n inliiin ami picsHiuc', n ahnuL 27 5 ^ 10 lft foi any <il ()" mid .1 pus-'UK ul one uimosplioie (KOO Jc'.ins 1 Di/mumml Tlnoi i/ of ('ft, p S) Tlicn ivoi'iffo (iHliuuo iipail, A\!UC!I ia |7 rr r IH tlioicfoio about one e niilhoiiHis of a coiitimelio The nuinboi of molcciil<"( pci inol is 22100 N or . x 10-' ho maRg 7>i of a nioloculo m any gns ma)' bo found by dividing fcho density by N Since the density of oxygen is 0011-29 ginmmo per c c tho mass of xys^n molecule is about 52 x 10-- 1 giamines The mass of a hydroprrn molecule ie-si\lecuth of tliat, 01 about 3 3 x 10~ 21 giamines, tlie latio of the molecular lits being 2 to 32 356 THERMODYNAMICS [ u'i' 221. Perfect and Imperfect Gases Thus I ho mola-nlai llic.i\, for gases which satisfy the tlucc poslnlalcs, gives lesiills idcnlic.i! with those we already know as laws of ideal pi i (<'<! i>.si s In a ical gas the postulates do not si i id Iy hold. Tlu si/i ol (In molecules is not negligible, and in any cnc-onnli'i I lu'ie is .in .ippn - ciable time dining which the molecules conevi m (I < vi I lmr(s on one anothei Theie may even bo lempoiaiy p.m m<> <>i cn-,inir.i- tion on the pait ol some molecules It is ink ie-,1 mt> In i IKJIIIIC, in aoeneial way, how these dcpai lint's lioin I he idi.il coiidihons all-l the calculation of the picssuic For this pin pose, considei Ihe simple case in winch one ol ,i gionp of molecules, advancing louauls lhc wall, meclsa molidili, initially at lest, to which it pisses on UK whoh ol ils inonu nluni, and the othei molecule then completes UK IOIIIMIN and ddmis the blow If theie weic no loss ol Lnnc in UK uicoimlu, and il I IK second molecule could be ie<>aided as havdlnm o\ ci < \adly llu lemamclei ol the distance, Ihe uile .il \\ln<li I lit' \\ill i-<i\<s momentum uonlcl be exacll> I lie same .is il I he c IK onnh i li.id nol taken place But if theie wcie loss ol lime in ,IIIN < ..... nnhi -IK h for example, as would ocein il Ihe hvo colliding niol(<iil(s mo\(l togethci foi any appiccuibk lime', uilh Hun \<lo(il\ xdiuid below that of the molecule \\hich \\.is onnin.ill\ imuin^ linn the late at which the \\,dl uti'i\ ts nionu nl inn \\ onlil l>< K diu \ d with the icsult of leducniii /' On UK olliu hand, il I In moliinhs have a finite size, so IhaL Ihe OIK \\huh \\as nnli.dh .il u^l had less distance to tia\cl in compk lino Hie IOIIIIIM , llu ial< ,il \\huli the impacts succeed one anolhei on Ihe \\all \\oiild Ix in<'i(as<d with the result of meicasmo /' This indicates thai Ihe pussine in a i< al i^as \\ ill dilh i lioin I IK ideal piessuie, which is ni\en by Ihe e(|iialion 1*1 ',,>' l>\ l\\o small teims, one posili\e, de peiulmg on Iliesi/t ol llu moh <'iiU s, and one negatne, depending on Ihcn eolusion. Such, in ilhrl, is the kind ol moelificati on \vlnch finds t \piession in chaiaclu ist u- equations like those ol Van dei \Vaals, C'lausuis, 01 C.illi ndai 222 Calculation of the Velocity of Mean Square Taking, loi any gas that may be tieated as sensibly pc ileel, I he i iinalion it is easy to calculate Ihe value of Ihe velocily ol mean sqimii' ,' when we know the density ol Ihe gas al a yive'ii pressmr The ill MOLECULAR THEORY OF GASES 357 pioclucl mN is the density, and we do noL need to know m or N scpuiatcly Lo find v In oxygen, for example, at C , the density is 001 1.20 giammc pci c c , when the pressme is one atmosphere, or 1 0133 x 10 G dynes pei sq. cm (Ait. 12) Hence in oxygen at , , , /3X l~0133x ]0 G , standaid LemperaUne andpicssmc, ens /* / --- nT^i --- ' ec l ual V W \ I 'J A. ^l l Lt J Lo JG1 mctics pci second Sinnlaily in niLiogcn it is 193 melrcs per second, and in hydiogen 1S30 metres per second 223. Internal Energy and Specific Heat. Consider next the beaimg of Lhc mokcnlai theory on the internal cneig} and specific heals ol a ^as We have seen that, in an ideal gas, wheic E' is Ihc encigy of lianslalion of tlic molecules, or ]>nNv z . This ma> be will ten KT = \E' 01 E' = \RT E' is llicitJoic piojioi lional lo the Icmpciahne Now E' may 01 ma\ not be Hie whole inleinal uitioy, E, which the nas acquius u lie ii il is lic'.iicd It will be (he whole li, when Ihc "as is heated, he molecule's can onl\ lake up enei^\ of tianslalion, and cannot ,ikc up (ii<io\ r ol lolalion 01 tncigv of \ibialion (\it 171). iupposi , loi instance, llial each molecule beluncd like a pci feet ly iiuoolh MUM! hilluid ball, 01 like ,i massix e point \\iLh no ajipicci- il)le moment ol meilia about an\ line passing lhioiii>h it In thai as( il could not ha\e aiu r encii|\ ol vibialiou, noi ,if<]uiie' nnv IK 1 1>\ ol i dl at ion in I he couisc ol ils e'licoiuile is \\ ilh olhei molc- uhs, ,iu(l llic oul\ kind ol e'oiiummie'.iltle kmelie eiuigy \\oulel je (iiein\ ol lianslalion \Ve should Ihen find /',' = /',", <inel con- (([iKiilly E - \RT \\'\\( n a gas ol this kind is heated, we' shoulel theiefoic have dK- \KdT iul in anv gas (icgaided as ])eil'ecl ) (IE - A',//'/ 1 anel A',, - K, -|- R lence I'oi a ^><is ^ hose molecules have energy oC lianslaLion only K = Ut K - r 'K JV u J > l> i ' K 5 ud y or -_^ = - 01 1 007 A .3 'his value of y would not apphy if E' wcic only a pait of E But is found that in a monalomic gas, such ds argon, or hclmm, or the 358 THERMODYNAMICS [APP vapour of meicuiy, the value of y is in lad cc[ii,'il lo "I 007 01 very near it The infeience is that in a monatomio gas, Ihc stmcluic of the molecule is such that substantially all its communicable energy consists of energy of translation In any gas each molecule possesses llucc dcgices ol freedom of translation, namely, freedom to move along cuch ol Lhiec inde- pendent axes. Since E' = ^RT, each ckgice of freedom ol lians- lation accounts foi a quanlity of kinetic energy equal to \RT This is tine whatever be the numbci ol aloms in I lie molecule, and whether or no the molecules have othei eueigv besides eneigy of tianslation. Consider next a diatomic gas, each molecule of \\lueh consists of two atoms. According to modem views 1 an atom is a eompUx system made up of a minute positn eh chaiged ccnhal nucleus in which the mass of the atom is almost all conccnliatcd, with electrically ncgati\c particles called elections elislribuled .uound it, at distances which arc large compared \\ilh tlu dune nsions of the nucleus [. The sliuctuic of the ,ilom and lhc naluic e>l Ihe forces between one atom and another m tlie 1 mole'eule au slill uncertain, but lor our present purpose it will suffice lo pie-hue an atom as a niassn e point, smiemndeel b\ a masshss quasi-claslie lender clue to forces which keep olhei aloms ,il a dislance Uuehi normal conditions a diatomic molecule is equi\al<ul, as regauls inertia, to two masses helel some dislance apail dynamically il may be compared to a dumb-LJI, a mou- e\ael compai ison would be to a light stick capable ol some elastic cxluision and can \mg a hcavj ball at each end Consideied <is a ngid body il has h\e effective degiees of freedom elfectue as icgauls UK slonug ,ind communicalion of kinetic cue igy name l\ r , I luce ol hanslal ion and two of rotation [ The two cfl'cclu e dcgic'es ol htedomol lolalioti arc about axes in a plane pcipcndieulai lo Uu x line joining Ihe 1 I wo atoms about that hue Usell, Ihe system has no elk el i\ e moment ol " Su ttiilliitlMiil, Pli 1 1 l/V//,M.i\ l')ll Bnln l >: nl Mini .Julv, St'pi and No\ l ( )U,,f J Tlioiuson Phil MIKI A|inl, !')!<) "I la an clcctiicilly iiculiiil alum lln< i)osi(ivc < l(u tin il v in (In inn IMI-J it (<|ii,il to the npy;,ilLvcclec!Liiut3 in (he I'lc'ctious Konmval ol ono m IIKIKMI! (In c'ldlnuiH would llioicfoic lcn\o tin. atom us ,i wlifilo |IOHI| ncily < lniij, f ( d I IIIH hiip|i( us wlion d ^is is ' iiuiiZL'd " \ AiiLC ii^id hcid\ luia si\ doniccs of IKUM|OIH il ( uu nio\c pniallit! lo itsell alony thioa iiidcpendenl a\oa, mid it mil iohil(^ about lhcs( avi's 'Vny poHsihh mo\emcnt i^ marie up of Ihcso MI\ components In a (lialoinn inoh uiln one of I lies degieos of freedom of lotation is inelloclivo us it)u;ai(N thu coiiiiunnicadoii of energy from one tuoleciile to anotliei in an cncountci ] MOLECULAR THEORY OF GASES 359 cilia Under these conditions it can be shown lhat Lhc ultimate suit of collisions is that the kinetic energy becomes equally laied by each of the five degiees of freedom The energy of anslation E' is equal to }RT, and each degree of freedom of anslation accounts foi an amount of eneigy equal to }RT It Hows that each of the two degiees of freedom of i otation accounts i addition foi }RT, and that the energy of translation and rotation >gclhci amounts to r [RT Hence if theic weic no sensible cncigy of ibiation as well, we should have the whole eneigy E = ~RT and K j, = r ',R, K v = 7 7 /t\ and y = | 01 14 Now in most diatomic gases, such as oxygen (O,) 3 mtiogcn (N 2 ), u, hydiogcn (II 2 ), nitnc oxide (NO), or caibomc oxide (CO), it in tact found that y is equal, very ncaily, to 1 -i at oidmaiy tem- ciatuies, and the mfcicncc is Lhat the stiuctuie of their molecules such as to give n\ c effective degiccs oi freedom, namely the fi\ e ml ha\e just been dcscnbcd, and that then molecules do not, at iduui v tcmpcialuics, hold anv consideiablc amount of commum- iblc ciK'igy m any othci fomi than as cncigv of tianslation and iK'igV ol jotalion But when such gases ate stiougl\ heated we no\\ lhal the specific heal mcicascs andy is i educed This means lat cneij>\ of Mbialion is then developed, which at high tcmpeia- IKS becomes an impoilanl pai t of the whole encigv In tiiatomic gases it mav be coii|cctuud that the iluce loins of anv molecule gioup lliemschcs not in one sliaight uc \\lueh would be an uiislable aiiangcmonl but so that le massive cuihcs he at Hie- cuineis of a tiiangle Sinulaily 'ht'ii Iheie 1 aie moie lhau Iluce atoms m the molecule, they 'ill place lhemsel\cs with llu u massive ecnhcs at the coincis I a polyhediou In any such Uiangulai 01 polyhcdial stiuetuie, onsidoied as a ngid system, thcic aie si\ effective degiees of eedom, nainelv thiec of lotation as well as thiee ol tians- ilion, loi theie is a liuitc moment of meili.i abonl anv axis, nd I he siiucture is such that Ihc molecule can be sel spinmug bout any axis by eneounteis wilh othei molecules As an ultimate ustilt of manv such encountcib, it may be shown lhal each of the hice degiees of freedom of lolation hikes up a shaic ol Ihc kinetic nergy equal to that of each of the tluee degiees of freedom of uuiblation, namely, \RT, and consequ cully that 1hc six degiees ogcthci account foi a total of SRT That is the energy which he molecules possess in virtue of their movements as rigid 360 THERMODYNAMICS [ \pp stiuclmes If tlieic weie no othei way in which Ihcy could take up energy when the gas is heated, we should consequently find, in a tuatomic 01 polyatomic gas, K = 37?, K p = itt, and y = \ or 1 333 The actual value of y, as expei imentally mcasuicd, in Lhc In- atomic gases C0 2 andII 2 0, is lathei less than Lhis, and in gases of more complex consLitution it is generally a good deal less IL is also found that the specific heals aic gicaler lhan 3/i* and \>R The mfeience is that m such gases the molecule geneially takes up a consideiable amount of cnerg} T oi vibiation m addition lo its cneigy of tianslation and lotation It appeals Lh.it a complex molecule can absoib energy not onl} by moving as a ngid body but by internal vibratory movements which ausc Ihiongh quasi-elastic defoimation of its own stiuctiire (Compaie Ai I 17 J ) The mam parl ol this encigy of vibiation piobably consists of to and fio jno\ cmcnts on the pait of the massi\e ceulies of Ihe linked atoms It is obvious that such a motion might ocem in any molecule that is made up ol moie than OIK atom Tin illect m a complex molecule is such as uould occm il the lines (oniing the massn e centies of the constitucnl alonis behaved like still spiings Thus m a diatomic molecule tvc might Hunk of the "dumb-bell" as haMng an clastic shank which allowed the distance between the h\o masses to ^alv The I'.icl th.il in a diatomic gas at oiehnaiv tempeiatmes the obscned specific heals arc <ippio\i- mately : lR and y?, and y is appioximaleh ] I, shous, Iie)\\c\ei, that the diatomic molecule then bcha\ es like a dumb-bell \\ilh a neaily mextensiblc shank BuL when Ihc tc'inpcialme is high, the vibiatoiy motion becomes iclalively moic nnpoit.inl, and il accounts loi an appicciablc pait of the whole encigy, c\ en in a diatomic molecule, and still moic in a liiatonuc 01 polyatomic molecule To this AVC must ascnbe the progiessixc mciease in specific heat, and the fall in y, which aic- obscived when any gas is heated that has t\vo 01 more alonis in the mokcule In a mona tonne gas Lhcie is no possibility of llns kind of vibui- tory motion, and theie is no cxpciimcnlal evidence of any change of specific heat vtith tempcratuic. The eiuigy depends only on motion of tianslation, and when the gas is heated its eneigy mci cases m simple piopoition to the Icmpeialuic I3ul when diatomic, tuatomic, 01 polyatomic gases aic shonglv heated, the eneigy mcieascs m a moie lapid latio than the lcm])eiature. This ij MOLECULAR THEORY OF GASES ,361 means thai the ratio of the total energy E to the energy ol trans- alion E' is not constant. In any gas that satisfies Lhe equation PV = RT, -, R y=l+ r . JX - U [f the total energy E presented a constant ratio to E', the specific icat would be constant, and in that case we should have y> constant ind equal to 1 + jE/E', since E, icckoned fiom the absolute zeio, s K T, and E' is \RT. The fact, howevci, that y falls, with using Lcmpciatiuc shows that the total encigy docs not picscrvc a con- tain 1 idtio lo Lhe eneigy of tianslation, and hence that theic is not equipaitition of the eneigy among the possible modes of motion In any gas we may wnte E=- E' + E" + E'". The cneigy of tianslation E' vanes as T, being equal to }RT. The eneigy of jotation E" bcais, in any given type of molecule, a n onslaut latio to E', and Iheicforc also vanes as T If the eneigy ol \ibiation E'" also boic a conslant ratio to E', Lhe \\hole encigy vsould Aaiy as 7\ \\luch is inconsistent with Lhe cxpeinnental (.Mills staled in Chap VI 224 Energy of Vibration The tcim E'" includes not onl\ .IK.'] yy due lo \ ihuilions ol the c onslilucnl atoms iclativcly lo one molhci \\illun Ihc 1 niokc'iik (//") but encig\ due lo \ibialions [IIION <. menls ol c Ice lions) willun I he consliluciit atoms LhcmscK es Jl a '") It is known lli.il l'] a '" is <i \eiy small i),uL of the whole iicioy, even at lempeuiliiics as high as 2000 C 1 The \ibialions hat m.ike up E tl '" have much highci fiequcncics than those that nake up A 1 ,,/" [I is lo \ ibiiitions willun Lhe consliluc.nl atoms that )iie alliibuLes Lhe bnghl lines which make up Lhe visible spcctium >l nn incandescent gas, and Lhe coiicsponchng daik lines due lo ibsoiplion in Lhe visible spec! i urn of light tiansmiltcd Ihiough i cold gas The longer-pcnod vibrations that make up E m '" cmil >r absorb rays which he in Lhe infra-red legion, beyond the range >L the visible spectrum. It is Ihesc longei -period vibiaLions thai 'onstiLutc Lhe mam part of Ihe vibiatory cncrg} r when a gas is .trongly heated, as in a llamc or an explosion, and give rise to nost of its radiant energy. Fiom the theory that has been outlined above, of the consti- 362 THERMODYNAMICS [APP. tut ion of a diatomic molecule, we should expect il lo li<i\ r c one well-maiked period of fiee \ ibiation, and thciclbic Lo show a stiong emission band when heated, 01 when excited by elect nc dischaigc in a vacuum tube, and also a coiiespondmg sliong absoipLion band when cold. A good example is furnished by caibonic oxide (CO), whose infia-ied spec ti um is found to consist almost entirely of one chaiacteiistic band, the wa~\ c-length of which is nbout 4 7 /JL ^\hen the gas is emitting uidiant cncig^, and I O^u, when the gas isabsoibmg it ! The fact that these wave-lengths aie so neaily the same is, evidence that what may be called the stiffness of the quasi-elastic link bctueen Lhe atoms, due to chemical affinity, suffeis little change when the gas passes liom the cold Lo the ladiant state Again, in the mfia-ied spectrum of the tnatomic gas C0 2 we should expect to find three piomment bands coiicspondmg lo the thiee modes of vibiation that can be scl up within a CO^ mole- cule by iclatne movements of the caibon and oxygen atoms) This is in agieement with \\l\at is obsu\cd Then- au, both in absoiption and emission, thiee distinct mfia-icd bands, namely a strong band whose wave-length is about I Ifj,, a weak band with a wa\ e-Iength of 2-7^, and anothei with a much longei \va\ e- length, between ll^and 15/j, (Ait 173) This long-pet iod \ibia- tion accounts foi the fact that c\ en at oidmaiy tempeialiius I he specific heat of C0 2 exceeds the value it Mould ba\ c 1 il theie weie no vibiatoiy cneigj, making y distinctly less than 1 ,W,*> Koi the pimeiple holds that \ibiations of long peiiod icqime no mou llian a compaiatn cly low tempeiatuic to excite them into taking iij) someconsideiable shaie of the cneig}', so that they then conliilmlc substantially to the specific heat, wheieas those ot shot I pcuocl do not begin to take up an appreciable sluue until the g.is is sliongly heated 225 Planck's Formula This pimciple Imds cvpussmn in a lormula de^ iscd by Max Planck to connect the oiurgv of any paiticnlar hequency of vibiation with Lhe hcqnency and wilh the tempciatme, when a .slate of cquilibiium has been icached * See W \V (Joblent^, Invrslirjutions of Iiifxi ml tfpu,lnt I'uhln ulioiiti of tho Carnegie Institute, Washington, No 30 1005, No dS, ]<)()(>, No !)7, 100S f N Bjeiuim, Vo> handlii ngc.n da dculsJicn Plu/t, GcsfllstJinJI, 1914, p 737, ilis ciib^e^ tlie lijpothetiral ronfiffuiation of a COj molecule \\hich would vibuito with penocls con expanding to the three obseived vAave lengths, whuli ho takos as 2 Ip, i 3fj., and 14 "/JL [i] MOLECULAR THEORY OF GASES 363 Accoidmg to Planck's theoiy the \ibiatory cneigy, per mol, coiiespondmg to any particulai frequency v is' 1 !,_ N ^ e RT 1 Hcic N l is the number of molecules per mol, namely G 10 x 10 23 , anel h is a constant, kno\\n as Planck's constant, which is the same for all gases and is appioximatel} equal to C 55 x 10~ 27 e^s R, as usual, is the gas-constant, whose value pei mol is 1 985theimal units 01 8,3 1 x 10 eigs, and e is the base of the Napieiian loganthms, "2 71828 The frequency v is equal to c/A, wheie c is the \elocity of light, or ,'3 x 1() 10 cms pci second, and A is the wave-length in cms. In a gas whose molecules aie capable of moie than one mode of vibration the whole vibiahonal eneigy E'" would be the sum ol as many lei ins, in Ihc above foim, as aie lequncd to expiess the \auons modes Thus in caibomc acid, for example, theic would be thiee tcims lot lieeiuencics e>f yibiation corrcspe)ndmg to the lluce obsened \\a\ e-leni>ths Vl any one dequeues v, lei the quantity NJiv/RT be upiesented bv ,(.' The n -j jff <ui(l Planck's loimula bcconu's \vheie ' is a laeloi the \ aluc. ol \vhieli depends on Ijoth v and T & { 1 lot any gi\en^ it lends to an uppei limit ol I when T is indefinitely mcieasedand lo a louei limil of /no when T is indtlimlcly leducerl Hence, if we accept I he loimula as valid, il follows lhat when I he molecule's of a gas aie dee lo vibialc in any one mode, the gas will lake up, m respect of I hat Uecclom, a quantity of eneigy \\bich appioMches the limit RT when the gas is sliongly he'alexl. This will l)e- I me also of .my olhei mode office vibiahon which I he molecules possess When Ihe gas is healed lo any given lempei aline the Jiaclion of RT which is laken up will in ocneral be difleienl foi different modes of vibialion, for il depends on the frequencv, being smallci when the ficcjuency is high This, aceoiding lo the Ihcoiy, is why the high-ficquency modes oJ vilnation Avhich aie ic\ca,le.d by the visible spec ti um do not contribute substantially to the * Foi a discussion of the tlicorotical basis of Planck's fonnula, see Joana' Dynamical Tkconj of Gases, Cliap XVIII 364, THERMODYNAMICS [AIM' whole eneigy of a gas, e\en at tcmpciaturcs such as ;nv teaelu (I m an oidmaiy flame 01 in a gas-engine explosion, ami why, in llu- leckonmg of eneigv and of specific heal il is only vibialums i>( mfia-ied frequency that need be taken into account. Koi (he same reason a gas whose molecules have one 01 moic loiiif-pmod lypes of vibration may, at oidmarv tcmpeiatuics, hold u considerable quantity of eneigy in the vibiatoiy foim, and have i spceilic lual markedly gieater than the ideal (vibialionless) vnlnr. The amount by which any one mode of Mbiahoii will augment the specific heat is found by diffcicnlialmg (-\\ilh ies|K-cl Io T) I he expiession for the extra internal cncigy lh.il is due- Io Mini mode We may wiite it ,77? f < ,,j ITS \ _ UiJl _ c _!! if A "~~ l 12 14 1C, Heie is a factor, depending on I he wave-length and I he tempeiatuie, which langcs fiom zeio Io unily as llic <|iianlil\ l/x is inci eased fiom zeio to infinity Fig, 90 exhibits (he maiiiK i in which this factoi incieases icUtively to I/a 1 . It shows thai llu ic is a veiy lapid use in the faclor, and Ihcicfore in I he speeilie heal, aftei l/x has i cached a value of aboul 1, bul up Io Hint pom! the effect of vibiation on the specific heat is quite insignificant. At C. the value of A \\hich coiicsponds to I/,/ 1 = 01 is OOOO.Ili, hence it is only those modes of vibiation whose wave-lengths are greater than say 5/z that sensibly affect Lhc specific lieal of a gas at normal tempeiatuie. As an example, take the diatomic gas CO with its eharaoleusf ic ] MOLECULAR THEORY OF GASES 365 bullion foi which A is about I- 7/i or 00017 cm. Foi that wave- nglh the value of I/a;, at C, is 09, and at 2000 C it is 74 ho factor c' t r 2 /(e l - I) 2 is theiefoie nisignificanLly small at C., it becomes about SO at 2000 C Hence the calculated specific \il K v , \vhich is }R at C , uses, as a consequence of this bialional cncigy, to ( r \ + 86) R at 2000 C , and the coircspond- i value of y falls from 1 l< to baiely 1-3. Again, take the tuaLomic gas C0 2 , one of whose chaiacleiistic ibialions has a, wave-length of nearly 15/x So slow a vibiation )ntiibulcs snbslantially to the specific heal even when Ihc gas is )ld At C a wave-length of 15/j, makes I/a 1 = 021 and e^ l2 /(e fl - I) 2 = 028. encc a single mode of ^ ibiation with that ficqnency should bung ic specific heat ol Ihc cold gas up to about (3 + 28) R, and cluce y horn 1 333 to 1 305 When the gas is stiongly heated, count has lo be taken of tluee modes of \ibiation whose wavc- nglhs aie long enough to be impoitant In itspeet of the thiec )gclhei, K obviously lends, at \ ei^ high lempeiatuies, to in- case to\vauK a linul of G/i', and y lo (all to ^, a,pai t liom an\ - ling thai olliei \ ibialions may conliibuie, and apail horn effects I dissociation Though I he ideas undeilvmg Planck's Lluoiv aic open to spulc, lh< ic c.in be hltk doubt that a, cui\ r ( moie <n less like lal ol li<> ')') docs u[)Kscnl the way in which molecular biiition ol a gi\ c'n l\pc eouli ibutes lo the specific IK at At si, when llu gas is being luale'el liom a cold stale, the conlii- ulion is piache.dh ml, MUM theic is a shai]) use, and finallv i as\ m[)lol ic appioach lowiiuls a limit The UmpciciliiK at Inch Hie sliaip use 1 begins dc [>e nels upon Ihc ficqnency ol liee ibi.ihon, being lugliei when Ihc dequeues is high The I'ael thai in pol\ atomic gases geneiallv the specific heals, al iinal li'mpc lahnc, aicgiculci lhaiitlu ielecLl(v ibuitionless) \ allies, id y is nolably less lhan 1 3.33, is to be asciibcel lo Ihcn [)ossessmg ug-peiioel modes of vibiation which aie icsponsive to low- mpeiatmc encouulers A complex polyatomic molecule may ive many such modes, cnch producing a substantial augmen la- mi of Ihc specific heal Smuiailv Ihc chaiactciistic mode of vibiation m a diatomic gas ay be so slow as lo affect the specific heal at noimal 01 com- nativcly low tcmperatuic, making K v gicatci than r \ll, and K v 366 THERMODYNAMICS | -vri' TI gieater than ^R, and y less than 1 i< This is nolably I he case the vapoins of the halogen elements C1 2 , Bi 2 , 1 2 These cleinenls have high atomic weight, and it would ,sccm lhal in each of them the pan of heavy atoms in the molecule, pcihaps lather loosely held together, have a slow type of vibialion, which explains the obseived high specific heats and low value of the latio y When a hydrogen atom is substituted for one of the pan, Ihis chaiaclei- istic disappear^, foi the gases HC1, IILh and III, when cold, <ue found to have specific heats that appioximate to tlic normal values, with the latio 1 4. 226. Effect of Extreme Cold on the Diatomic Molecules of Hydrogen. It has been found that Avhcn hydiogen is cooled to about -200 C. its specific heat falls piogiessncly to n, value not much gieatei than that foi a monatomic gas, and y uses to a value not much short of that foi a monatomic gas (1 667). This lemaikable lesnlt, fiist obseu ed m mcasuicmcnts of K a , has been confhmed by independent measuiements of K and of y ' . It appeals theiefoie that under extiemc cold the hydiogen molecule tends to assume a diffeient stiuctme, becoming in effect quasi- monatomic, piesumabh by the coalescence of the tuo atoms which, at oidmaiy tcmpeiatuies, aie held apait The pan ol atoms appaienth behave as if the foiees \\luch usii<ill\ liolcl UK ma pa it A\hat we called then tender in Ait 223 cease to be ellecinc m pieA entmg the massive nuclei fiom coming togethei, to loim what is vntually a single-atom molecule ol double mas--, It may bo conjectmed that this happens Avheu tlic lotational speed ol I lie diatomic molecule tails belo\\ a ccitam limil, ami llial tlu moltoule thenietams the coalesced state until its cons ti hie uL atoms aic loieeel apait by a snlhcicntlv \'iolent cncountci \\iule it umaiiis m I he quasi-monatomic state it takes up cneig^ ol liansl.ition onl\, and when a laige piopoition of the molecules aic m lhal slale I lie gas beh^ es appioximately as a monatomic gas in icspect of ils specific heatb So fai as is known this action is peculiai lo hydiogen, it does not occui in oxygen, mtiogcn, 01 caibomc oxide * Eucken, tiitzunysbenchte d L Pieuss Akad , Boilin Fcb 1012, School and Heuse, do, Jan 1913, also Ann d Phyt>il, Vol 10, p 473, 1<)1J, M SJuelda, Phijs Review, Nov 1917 APPENDIX III 'FABLES OF THE PROPERTIES OF STEAM ihlc A Piopcilies of Sal mated Steam, in i elation to the Tcmpeialmc A 1 . Piopeitics of Walei a I Sat i nation Picssuic. 1? Piopulies ol Salmatul Slcam, in iclalion to the Piessme Volume ol Sleam in am Div Slate Tolal Heat ol Steam in any Div Stale K Knhopv ol Slcam in any Diy Stale I' 1 Spctilie Ile.il, al conslant picssiue, ol Skain in any I)i\ Stale Tin sc Table -1 an based on Callciulai\ i'oiinnlas 3 and will scnc illii^lial( Ins mi I hods Tin (ionics aie, lot the most pait, taken run Thf Ctil/cncldi Sttmn Tahiti published b\ Ed\\anl Ainold, )l r ), \vlu< h will be loimd lo ni\e much moie complete paiticulaii 368 THERMODYNAMICS [APP, TABLE A Piopotiet, oj Sal in tiled Temp Cent Pi 03311 10, pounds pei sq inch Volume, cub ft pei Ib V, Tol.il Heat L Ib calo- nos pei Ib hltlOp\ , pei II) LfiU.nl Jloat, II) culn- IICN poi II) L Inlonul 11) (illo- uis pi i Hi 00892 3275 9 594 27 I 17602 59127 561 21 10 1788 1693S 59901 1 > 11 OH) 589 ? 507 85 20 03399 922 19 603 72 : 20l>221 58 ) 78 571 48 30 00162 525 81 608 40 > 01247 578 19 575 07 40 10703 31245 01304 %088 573 15 57804 50 1 7888 19272 0170') 92490 567 75 5S2 17 60 28873 12291 022 10 88021 562 29 585 00 70 4 5156 SO 804 020 00 85039 556 72 589 07 80 6 8627 54 596 030 95 81712 551 05 5<>2 4 1 90 10 101 37 815 035 19 78019 5 15 25 595 07 100 14089 26 789 039 30 75732 539 SO 598 S3 110 20777 19370 64/20 73027 533 17 001 80 120 28 SOS 14271 647 07 70-185 520 85 001 78 130 39213 10 090 (.50 72 08092 520 i2 007 58 140 52 482 8 1431 054 19 65S3I 5 1 J 57 01023 150 69 150 6 2895 657 47 0368'! 500 50 01273 160 89800 4 9232 660 55 01057 499 29 01508 170 11506 3 9015 663 44 5972 1 191 75 01727 180 14559 3 1275 066 14 1 57884 183 <3 01930 190 18208 2 5339 608 05 50128 47582 02 J 19 200 225 24 2 0738 670 96 1 51453 407 1 1 022 91 210 275 78 17134 073 09 52851 458 09 02^1 18 220 334 38 1 4285 675 00 J 51320 119 09 (.25 03 230 40189 1 2007, 676 87 19868 J40 38 027 23 240 47874 1 0178 678 55 1 I 4S480 43081 028 43 250 56563 8695 680 12 J L 4710 1 (20 90 029 53 TABLES OF THE PROPERTIES OF STEAM 3G9 Piopeiliei, oj IVaio at Salutation nip ont PtC-HIILO, pounds poi Volume, cub ft poi 11) Total Heat, Ib -calouea pei Ib Enliopy, pei Ib Function G, Ib caloiica pei Ib J'. =1>* v* /, <*u=-G* 00892 01002 10 1788 001003 998 03585 0181 20 03390 001005 1091 07040 0714 30 00102 001009 2991 10393 J 58 40 10703 001014 39 89 13031 278 50 1 7888 001021 4088 10770 4 30 60 2 8873 0011)29 59 S7 19815 13 70 45150 OOlolS 0088 22774 826 80 8027 001048 70 90 25052 10 OS 90 10 101 01059 80 94 02S151 1338 00 I I 089 001071 100 00 3118ft 10 30 10 20 777 001081 11000 33853 10 00 20 28 80S 001008 120 22 30400 21 10 30 39213 001713 13040 030011 20 81) 40 52 482 001729 140 02 041511 30 SO 50 00 150 001740 1500] (301)} 3 r > 10 60 89 800 001705 10! 20 10 {7$ 30 5S 70 1 15 00 001785 17! 00 U487H 44 2<) 80 145 59 001807 18221 051078 19 22 90 182 OS 001831 192 83 053381 5 I 38 00 225 24 001850 203 55 55054 50 75 10 275 78 001885 214 40 57004 O r > 13 20 3 J t 38 001014 225 37 00 1 28 71 12 30 401 80 001940 230 49 023 12 77 11 40 478 74 001980 21774 004517 8330 50 505 03 002010 259 10 OOOS7 89 OS THERMODYNAMICS [APP TABLE B of Katiiifitid 1 Piessuic, pounds Temp Volume, cub ft 1 Total Heat, Knliops L.Lll 111 limit, ' h'lmUun (1, pei &q Cent 1">AI ])"> 11) calnuos pei Ib III C ll()l 11 S Hi t,il<> inch t IJlsL Lt~r V pci Ib ^ poi 11) 1 11 S ] 1C 1 [ P ' & I* L C/,, _ (,' w 01 1 59 2940 595 03 2 J662 593 4 1 005 0-2 J 1 09 1524 599 81 2 1068 5S8 15 02-16 0-3 1799 103S 602 77 2 0727 58 t 83 058 0-4 2206 7907 604 97 20182 582 3b 091 0'5 2641 0505 606 73 2 0299 5SO 10 1 23 1 3874 333 J 61246 1 '172-1 573 8 ! 261 2 52 27 1735 618 67 1 ( H5 ( ) 56(> 52 1 69 3 6083 1186 62253 I 8S } ! 561 Si (. 30 4 0723 9054 625 38 I 81)00 558 28 701 5 7238 7344 627 1 I SI 22 555 !S 881 6 7672 619] b2<) 52 ] 8277 552 92 <) SI) 7 8049 53 59 631 15 I 8 11I> 55071. 1081 8 8384 4730 632 57 1 8019 5 18 82 11 69 9 8684 4230 033 85 1 7951) 547 0< 12 50 10 8958 3839 635 01 1 7S71 515 50 13 2(. 12 94 14 3237 037 02 1 7731 542 61 I 1 07 14 986(3 2802 638 77 I 7611 510 12 15 'II 16 102 11 2473 640 26 1 7506 5 57 S ! 17 12 18 103 79 22 16 641 60 I 7111 5 {5 7 r > IS 20 20 10887 2008 6L282 1 73 !3 533 S7 l')22 22 11171 1837 64392 1 72 5S 532 0') 20 Ih 24 11434 16 93 GU 93 1 7 1 89 550 1 1 21 09 26 11080 1571 15 85 1 712b 528 SS 21 ( > r > 28 11911 1460 04071 ] 7069 5J7 J2 22 7S 30 12128 1374 647 5 1 17016 526 02 23 51. 32 123 35 1294 618 JO 1 6<I66 52 1 67 2 1 33 34 12531 12 22 649 02 ] 6919 523 10 25 07 36 127 17 1159 649 69 1 687 J 522 17 25 77 38 12896 1102 65031 1 6831 521 00 21) 15 40 13007 1050 650 95 1 6792 51987 27 12 42 13231 1003 651 53 1 6751- 518 77 27 76 44 133 89 9003 652 OS 1 6719 51771 28 40 46 13541 9212 652 61 1 0685 516 68 29 00 48 13688 8853 053 12 16651 515 69 29 59 TI TABLES OF THE PROPERTIES OF STEAM 371 (coHltnut'il) of tiatiuatcd Stcant pounds Te nip Volume 1 , i ii It li Heat, l <1 ntiop\ , Latonl HcMl, KlIIKllOll put Hq Cent L> 1 1 f J L (j 11 11) lalo pin Ib 11) i-nlcim* i Ib Gciloucs moll t poi II) MLS pen Ib p(?T 11) poi Ib l> ^ I* J ('i = flw 50 138 30 S 520 053 00 1 0020 514 71 3010 60 1 14 79 7 JSt 055 77 1 0479 51022 3285 70 15046 (5218 057 (31 1 0359 500 23 3530 80 155 52 5 187 059 20 1 0250 502 59 3754 90 10009 4 913 000 59 1 0105 49924 39 02 100 1(51-28 1 451 001 82 1 0082 496 11 41 58 110 108 15 4 070 002 93 1 0007 493 18 4340 120 171 75 3 75 1 0'03 92 1 5938 49040 45 13 130 175 13 3 179 0(54 S3 1 5875 487 70 4078 140 17831 '3 245 005 09 1 5818 1-8527 4837 150 181 51 ! 041 oor. 49 1 571)5 182 90 4989 160 IS! 10 2 S02 007 22 1 5715 48001 51 34 170 ISO 88 1 70 { 007 90 1 5000 178 40 52 75 180 1S9 IS 2 502 (.OS 53 I 5020 170 20 54 10 190 191 97 2 135 (.09 J 3 1 5577 17 1 20 55 12 200 i9rr> 2 '!20 009 <><) 1 5538 472 21 50 09 210 I ( M> hi) 2210 07020 1 5502 470 20 57 94 220 I9SS7 2 120 070 70 1 540". 108 38 59 13 230 20! 02 2 54 1)71 19 1 5129 10055 0031 240 20'3 <)<> 1 95 1- 071 04 1 53% 40170 01 45 250 205 10 I SSO 072 07 1 5302 10300 02 5S 260 207 01 1 SI 1 072 IS I r . \ \1 401 30 03 00 270 21 IS 9 { 1 7 IS 072 SS \ 5 JO i 459 05 04 72 280 210 77 I OS') 073 25 1 527 t 158 02 (.5 77 290 21257 1 Oil 07'} 01 1 52 10 150 11 00 79 300 2 1 \. 32 I 5S5 073 90 I 5219 151 St 07 SO 350 222 15 1 308 075 52 I 500(5 11741 7257 ! 400 229 75 I 200 070 8 ( 1 49!)l 410(53 70 9(5 450 23(5 12 I 07') (.77 97 1 1897 43 1 28 810(5 500 2 12 57 0977 (578 97 1 J-8H 428 31 SI 92 242 372 THERMODYNAMICS vrr. TABLE C Volume, in cubic jeet pet U> Pieasiue in pounds poi sq inch Temp Cent 20 40 60 80 100 120 140 400 35988 17973 11967 89618 7 1632 60009 5 10-13 350 33 295 16617 11058 82785 66107 5 1989 | 1 701-8 300 30 594 15 254 10 141 7 5848 6 0509 5 0284 1 2980 290 30 052 14981 9 9569 7 4449 5 9 {78 JfJUO 12153 280 29 510 14706 97718 73045 58241 48372 4 1323 270 28 967 14431 9586? 71630 57100 47109 10-187 260 28 425 14 156 94002 70221 5 5953 4 (>! II 3 9040 250 27881 13 880 9 2134 6 8799 5 4708 45105 .58798 240 27 337 13 603 90260 6 7370 530^7 4 1 KSI 3 7<H2 230 26791 13 326 8 837i > 6 'V)33 5 2-1 Ii7 4 5190 3 7078 220 26 240 13 048 8 6483 6 4480 5 1289 -1 2190 '! 0205 210 25 699 12768 8 4582 6303L 50101 1 MM 3 5324 200 25 150 12488 8 266S 6 15()|. t8')OI 1 0159 3 mo 190 24601 12206 80743 ( 0085 I 7690 !'II27 ! 3524 180 24 050 1 1 923 7 8805 5 85 l >2 4 61 65 3SJ80 { 2000 170 23 497 11 638 7 6350 5 7083 4 5J2 t 37!I7 3 1070 160 22944 11352 74878 55558 1 4 3%0 i 02 {8 30718 150 22388 11063 1 1 72886 | 54012 -1 2687 3 r> 1 !8 29715 140 21 829 10 773 70871 52413 1 1380 3 IU|(, 2 875 1 130 21 268 10479 6 8832 50850 J-OOOL 32809 27731 120 20 705 10 183 66762 4 9227 3 8706 3 1791 2 0081 110 20 138 9SSJO 64661 4 7572 37J1S 30182 25599 100 19 567 95809 6 2522 4 5878 i 3 5892 29235 2 1180 ] TABLES OF THE PROPERTIES OF STEAM 373 of Si cam ui any Dnj State. 111 pounds poi si| inc.li snip 'out 160 180 200 250 300 350 400 100 350 (- 4()() f ) 1 1001 .'{ flfiOf) > (>t59 3 5I>01 .i 27r>j 2 8,wr> 2(>08l 2 JflOl 2 K.IJ'j 20 189 1 SJ58 I 7586 1 b075 300 290 280 270 260 '{7,101 , (>77I , ()()}(> r>2' n l r ), r )() .5 :5240 j 258 r ) ,j l')2l :: 1 2 is >05S7 2'KSil 2 02,ir 2 S(. !4 28027 2 7 M(> 2 ]()<)(> 2 J20I, 2 2712 2 2 '1 2 2 170') XiOG <)IS7 S7(>5 S i,'57 7,V)4 1 (>(iS,{ 1 (>,]17 i r ( )4r> 1 r ). r )(i8 1 JllSI) 1 44!)2 1 4K.4 1 JS'50 1 .'1401 I ^18 250 240 230 ,57'i7 , 5057 , 2200 2 0008 2 '22 2 h.-20 2 (7 l l7 '2 (.171 2r>fi!i 2 I I'll) 201)77 201 t'l 7I(J 70 l r > () r )") ( ) 1 4701) 1 HJ!I<) 1 Wi 1 2701. 1 2M7 1 2071 220 , 1 I'U 2 7S2<> 24VH 1 i!4 1 I)0'I4 i ir.so 1 100~) 210 j 070(i 2 7 1 1 -1 2 1211 1 'HlthS ' 1 r .(.20 I iir>8 1 1 !IO 200 2 0008 2 (){'() 2 J. r j7( 1 S r ,l 1 " 1 ill f. 1 272 { 1 001 I 190 2 ( )0')7 2 5<>5 r > 2 2 f H)it 1 7'MI 1 H!7 1 2271) 1 OHOl 180 2 h27l 2 lOOli ) j > |o 1 7 :i>0 1 II2(. 1 ISK) 1 008! 170 27(51 2 mo 2 noi 1 <>7f>() 1 r>')S i i t,:<) Olil r > 160 _> I)T7S 2 i.ns 2 07S2 1 dl l r > 1 I0 r )l 1 OMI> 00100 150 2 r 70l 2 2'~>5 r ) 2 00!0 I r, r (is 1 218') 1 O.H2 08714 140 130 120 110 2 IS02 2 W7S 22021 2 1057 2 17.51 20SSI 2 0001 I DOS') 1 <)27 5 1 SIS! I 7((. ! 1 <>SIO 1 IS'll 1 II (.7 I nr>i 1 270S 1 l')02 1 I2'IO 1 01) l<) ( )')7r> 0'I7<)() o <)2:ir> os()ir> S022 S2 1 7 071)01 071 M <) r ) r >7 100 2001:5 1 SI 10 1 5 ( )2() 1 I')2(> 920 ! 07l()I 50,54 1 374 THERMODYNAMICS [AIM 1 TABLE D Total Heat /, in II) -colon es pet Ib Picsauic in pounds poi s<| nu'li Temp Cent 20 78470 40 60 80 100 120 7S2 23 110 781 71 400 78420 78371 78322 78273 350 760 G8 76004 75939 75874 75810 75715 7f><> SI 300 736 61 735 74 734 88 73 t 01 7 ! ! 15 7 52 2S 75112 290 73178 73086 72994 72902 72810 727 IS 721. Jd 280 72695 72597 72499 72402 72301 722 0<> 721 OS 270 72211 72107 72003 71899 71795 710 01 7IOS7 260 71727 71016 71506 71305 71281 71173 7I<>(>2 250 71243 71124 710 Od 708 S'7 707 (.0 706 51 71 H !2 240 70757 70631 705 04 703 7S 702 ".1 701 21 (,(,() 4)1) 230 70271 701 36 700 01 608 ()(. d07 !0 (.01 <I5 d<> / <)() 220 69785 606 40 691 05 (.0 3 50 (.02 05 600 (.0 dSM Id 210 69297 691 42 680 86 (.8S !l dSd7d (.85 20 dS ! (>-> 200 688 08 686 42 684 75 (.83 08 (.Si II (>70 71 (.7S OS 190 683 19 681 30 679 60 (.77 SI <>7dOI (.71 22 (.72 1 ! 180 678 28 676 35 (.7141 d72 48 07055 66S 62 ()(>(. (.0 170 673 35 671 27 669 10 6(.7 10 dd r . 02 ddJ ')! ~ddO M" 160 66841 666 Id (.6391 661 66 650 II (.57 Id (.5 1 Ml 150 140 130 120 110 100 663 46 65848 65348 64846 64340 6,38 31 661 02 65584 ~ 650 61 645 34 64000 63459 658 59 (>5d 15 650 56 6-1 I 88 6'39 00 63'3 10 627 15 61'! 71 61702 61201 6 !5 07 620 79 623 i,{ dll 2S (.I52S d.i!) 1 1 (. !2 S5 626 3S (.1971 (.IS SI 6!<> "21 dir> ')' 653 20 6 17 75 6-1221 636 50 630 87 TABLES OF THE PROPERTIES OF STEAM 375 aj fitiani in a in/ Dnj Stale J'ICV-HIIO m pounds pei srj inch 'cm]) 'cnl 160 180 200 250 300 350 400 400 78 1 2,1 780 71. 780 2<> 77!) 0,} 111 80 77(i 57 77.1 U.J 350 7fid Id 7 15 ill 71487 7.13 25 7,11 di 750 02 748 10 300 7 50 ill 720 dO 728 82 72d dd 724 .10 722 .5 t 72(1 17 290 721 ,14 721 I,} 72. if. 1 72 1 2 1 7 IS 0] 7lddl 7lf Jl 280 720 1 1 710 1 1 7IS 1,1 7 1 ,1 7 1 71 i 2d 71082 70S 57 270 7 1 1 S ! 71 ! 70 71271 710 14 707 11 704 01 702 >4 260 700 11 70S 10 707 20 701 12 701 7.1 dOh OS (.0(. 21 250 7<H U 702 01 701 77 I.OSSI (.0.1 hi d02 SO dSO j 240 (.OS 72 (.07 Id dOd 10 dO { ll.{ I.S'O S7 dS'd 70 dS i 14 230 (.0 i 2t dOl SO dOM,14 dS7 Id (..S i T7 dS'O 10 (.77 01 220 <S7 71 dSd 2l> dM SI LSI l<) (.77 r.d | 1.7 JOJ d70 ,!2" 210 dS2 10 dSO .U d7s 0<) 1)7". 10 (.71 i > dd7 !l dd! 11 200 190 (>7d 1 1 d70 (>.] (.71 71 ddb SI -H'-!f- ddS OO d(>2 Id ddl dlS OS ddO f>d did .50 MM II 180 (id 1 7d dd2 SI did 07 dll 21 Old II dll IS 170 dlS 77 did (.0 d,l 1 dl) 1.10 10 dl 1 10 d {S OS di! 77 160 (,12 dd (.10 II dlS Id (.12 l'{ d id 01 (ill 2S (.2;! dd 150 did II (>H 07 dll 1{ d i! 14 (>20 ,!,! I.2.J 2<> (.17 17 140 dlOOO (.17 .!(. d.il 72 (>2S 12 (>2I 12 (.1 1 02 dOS ,!2 130 (. i ! II (.,!() 11 (.27 1.7 (i20 10 (.I.! n 5 dOd Id 508 00 120 (i2d dl (.2.! IS Ii20 id (.12 Id (.01 71 lOd 01 .ISO 1.1 110 dlO .17 did 17 d!2 77 (.01 2d 101 71 1H7 "2 1 57.S 7.} 100 (.1227 dOS 11 d(M S { ,10.1 ,12 5Sd 22 17d 02 ,ld7 (.2 37(5 THERMODYNAMICS TABLE E. Enhopy </> of Steam in pounds poi sq inch Temp Cent 20 40 60 80 100 120 140 400 20100 1 9331 1 SS78 1 8556 1 S304 1 8097 , 1 7921 350 1 9729 18958 1 8503 18178 1 7924 1 77H J 7530 300 1 032b 1 8551 1 8092 1 7764 I 7506 1 7293 1 71 Jl 290 1 9242 1 8465 1 8006 I 7676 1 7117 1 7203 1 7021 280 I 0155 1 837S 1 7917 1 7586 1 7327 1 7112 1 6928 270 1 9067 1 8288 1 7826 1 7494 1 7234 I 7018 1 6S33 260 1 8977 1 8197 17734 1 7401 17139 I ()92I 1 6735 250 1 8885 1 8104 1 7639 1 7305 1 7U41 1 0822 1 (.635 240 1 8791 1 8009 1 7543 1 7206 1 694 1 1 0721 1 (,532 230 1 8()9G 1 7911 ' 1 7413 17105 1 6839 I 1)1)17 1 6*26 1 220 1 8598 1 7812 1 7342 1 7002 1 6733 I ()5U f ) 1 6316 210 1 8498 17709 1 7238 1 6896 1 0025 1 ( J98 I 020 1 200 1 8396 1 7605 1 7131 1 678(5 I 6513 11.281 [ OOS7 190 1 8291 1 7497 1 702] 1 0673 I 6397 I 6161) 1 5' 106 180 1 8184 1 7387 1 6907 1 6557 I 1)278 1 604 1 15811 1 170 1 8074 17274 1 6791 1 6 1,37 1 6155 | J 5917 1 5711 160 1 7961 1 7157 1 6670 1 6313 ~1 6027"' 1 .->78 r ) 1 5575 150 1 7845 1 7037 1 6546 | 1 6184 15894 156 IS 151-33 140 J 7726 16913 ___ 1 6050 J 5755 1 550 1 1 5285 130 1 7604 1 6785 1 6283 15911 15610 15354 15129 120 1 7478 1 6652 1 6144 1 5766 15458 15196 11901 110 1 7347 1 6315 1 5999 1 5613 ] 5299 1 5029 1 1700 100 1 7213 16372 1 5848 15454 151 31 14852 14605 II TABLES OF THE PROPERTIES OF STEAM 377 in any Dnj Stale. Temp Cent Ptessuio m pounds pei sc[ inch 160 ] 770S 180 200 1 7511 250 300 350 400 400 1 7033 1 7250 1 7034 1 0849 I 0087 350 J 7381 J 724.) I 7118 1 0852 1 0030 I 0439 I 0271 300 0952 I 0810 0082 1 0400 J 0175 1 5970 1 579S 290 0801 1 0718 0589 1 0311 I 0077 1 5873 1 5090 280 0707 1 0023 0193 1 0212 1 5970 J 5771 1 5589 270 0070 1 0525 0)91 I Oil! 1 5872 1 5004 I 5179 260 0~>72 1 0425 029 J 1 0000 1 5704 1 5553 1 5305 250 0170 1 0)22 0188 1 5898 1 5052 1 5438 I 52 10 240 0)05 1 02 Id 0081 1 5780 1 5537 1 5310 1 5123 230 0277 1 0100 5909 5071 15117 1 5194 1 4995 220 1 01 10 1 599 ) 1 5854 5551 1 5292 || I 5005 I 4800 210 200 190 1 0031 1 5912 1 5789 1 5S70 1 5755 5029 1 5735 15011 1 5182 " 5420 1 5102 I 5020 1 1881 1 478S 1 1719 J 1571 1 4110 5290 5101 180^ 1 5001 5197 I 5 548 50 M) 1 1735 44S2 1 4251 170 1 5527 5500 J 5208 1870 1 1577 J )IO J 4077 160 I 5387 52 1 7 1 5000 1713 1 Nil 41 10 1 3891 150 1 5211 5007 1 4000 1 1518 I 12') 5 '!95 3 1 309 ! 140 1 5088 4908 1 4743 1 1373 , I 1047 3753 1 3482 130 1 1920 1741 1 4570 I 1180 I 38 17 353!) ] 325 ) 120 1 4755 I 451) 1 1 4380 1 3980 1 3031 I )307 ] 3000 110 1 1571. I 1375 1 1190 1 3772 1 3')99 1 3050 ] 2737 100 I 1381 1 U74 1 3980 J 3511 I 3147 1 2783 1 2413 378 THERMODYNAMICS LAPP TI [ CD <M 00 1C 1C IO 1C 1C 1O 1C <N C3 O O 00 00 C5 <N -H l> 1C 1C O O O O O >C -H r- l> ic CD co o on ic ic ic ic o o co O CO l> r-H CD (N O O O 1-1 i (M 3 1C 1C 1C 1C 1C O 00 O-i 01 00 CO 0| (M <-3 -H O I- Cl 0| 1C 1C 1C 1O 1C in O O i I -rh CO O1 00 oo O O O > i I 4 1C 1C 'C lOi 1C -H 01 o-i co co 10 co iM i"A -fi in O CO O in >o 10 >c ic m o o oo i ( -H co f3 cr -s >c ic r-~ i~ 2 -H ic m ic ic m in ic ic >c i" ic co > i in c: -t<oi--^co r i c 1 t-- o -ti oo <"^ c- t-- i-- cr 11 ClO ~ O - i i 01 ii-^-C -H in in m ir i- u- in in in _ jO C5 i I ~3 lit CO ni o O 1^ 10 10 CO -H cooccicrcr c o o i i i < 1-1-^-^,0 co ~p -ti -H -t -ti -f in IP IP in in 10 in in O uOCOi-OOOCl CjC^dOO ' i ^ I rn H D -fi -t 1 -ti -t -ti -t -t -H m ic >C 101 in ic fa H iMco-tiino coooi-fji^ i i^oi~c O COCOOOt-OCO COCjCS Oi O^ O O i i o | ro * -f -+ -t< -* -f -ti -H -H -h -H ic in ir in in O O i < i i O-I n-ic^-FOIr- CS o oooocoooco cooocoooco oo T- OOOOO OOOOO OOOOOO Ojr 1 OOOCD^Ol O 00 CD "^ CXI OOOCO^CMO tfcocococo COCSJCMCXICXI CM-H-H-H-MTH INDEX J)Holuto /oio, 12, 35 Lbsoiption bands in spectmm, 3(51 Ldiabatie elasticity, 2Sb expansion, 21 elloci on di\ness, SI ol a fluid, 81 ol a peifoctc;a3, 22 of slofiin, 328 idi.itlioimal piocesfl, 129 i" ' 'I' 264 k, of, 59 ui, isotlicimals of, 302 sopnialionot thoconshdit ntsof 180 stdiiddid, 229 111 UltnilP, HI^CIK ItlLlVC, 5J Jouli '4, 5 r > Vn m,i< Innos Em i cl 1 141 Ml ion, 1 "is ullvi n, SI, 'm im.iual, K If , 200, 302, 310 iI2 HIlclL'cllN ISOtllC Illials, 2')') nuindmir to ( '.il- li mini s equation ot siati , 12 5 iiimioma ilis<n plion in.ifliiiK, 11)1 d.iln loi, 1 T) MSI ol 111 IcflL!{Uil.(l<H), l,}7 LnllH^^^'^ ill) ilLinson 2 r >(i ilinnsplu 10 pirssuio of Rlandaid, S ilonni sd uc tmo, 3 r )S i\of,' idio's ld\\ , 235, 355 , mean, 331 C'aiboiuc acid, cidical teinpciatuie of,SO claLaloi, 145 i&otheimal for, according to Glauauis, 317 isothoiinnls of, 300 inolccnlai vibi alien in, 302 specific heat of 245, 304 use of, in lefngciation, 138 Garnot's cycle, conservation of entropy in, 46 ot opeiaticma, 27 icvoraod in lotugoiatmg inacliine, 138 \sith a jtcitect gun foi \\oiUnifj; biiliHt.into, 30 with slcani foi working siibslaiKo, S8 ('(HiiKlo inrlhod ot liqui f\ iiiL,' pases, IG'l Cli.iiiii Ii iistic i qiialion, 2 ( )0 Callondai's, 31 Clan <nis\ 315 [)K tcuci's, 317 of n )>(. i fee t ^nfl, 200 Van dei \Yauls 1 , lily's cm v( i, 185 i( an do Jvochas, 22f> 5oll ' i , 5|i n ' ! ' Join, N , S5S ?0lllllr, 310 iniindai y cuivos 02 !.i\lo's l.uv 13, 355 SmiU'otUi, 320 Clinks 1 la\\ 1 5 (hail of utliopv and (i inpc lalim , 118 - lolal heal, 121 ( hails ol pmpcidi's of Hinds, slopes of lini d in, 2SO ( 'he inical i ontnu lion, 23(> machine, 15H Cla})eyion's equation, 115,283 application of, to i linngcH ol fllalo, 1 1 (> Cldiido'n aj)|)aialiis loi complete jcctili ( uf ion, 188 liqiu ladion of an, 178 Sulish Association Committee on (,'laiisms, 50, 315 gaseous ovploaions, 2 12, 2 I I, 241, 251 , ( 'hmsm.s c haiactcii d ie equal ion, .515 257, 2(55 CloiiK-nt and Dcaoiinos, 204 Jntish ThennalUnit, 9 r ' " r- '' '^ 229,244,251,259 5nbblos, cquihbiiuin of, 350 i ' Juckingham, E , 277 ' .1,1 3, 319 in alcani, 32!) 'allendai, IT L , 12, 70, 72, 73, 70, 83, Coblcnt/, W W , %2 102, 108, 127, 201, 257, 208, 315, 329, Ooofhcicnl of poifonnanLO, 2, 134, 13G 335, 338, 350, 307 Coffoy still, 181 'allondai and Nicholson, 207 Collected " ~ r ^ " r relations, 287 'allondai's cliaiactonstic equation, 318 Combustic steam tables, 63, 340, 3(j7 Compound i ' I ! 220 380 INDEX Compressibility of a fluid, 2S6 Compression, advantage of, in gas- eu "ines, 231 Condensation in expanding steam, 207 of water-vapour in an, 07 Conservation of entropy in Cainol's cycle, 45 m lefncei- atmg piocesa, 135 Continuity of state, 304 C'onveigent divergent nozzle, 194 Cooling effect of tlnottlmg, 127, 270, 296, 314, 318 Con espon ding states, 311, 317 Co- volume, 307, 319 m steam, 328 Cntical point 80, 304, 309 in steam, 320 on the I(j> chart, 123 T</> chait, 120 temperature of cat home acid, 80 of hvdrogen, 81 of steam SO Cuihs steam turbine, 220 Curve of condition 218 Cut oft, 97 Cycle of operations, 4 Dalby, W E , 125 2,57 Dalton's principle, 05, 355 David, W T 200 Decrees of fieedom m gas molecules, 201 358 De Laval's nozzle, 193 steam tin bine, 215 220 Diatomic gases 261, 358 specific heats of, 240 359 molecules of h\diOLrcn, effect of extreme cold, 505 Diesel engine, 234 Dietoiici's characteiibtic equation, 317 Discharge through nozzle, 194 01 ifice, limitation of, 197 Dissipation ot eneiLry, 50 Disbocratron, 204 Drops, foimation of, in supersaturated vapom, 85 344 Diving air bv apphration of cold 07 Drying pipes 100 Drvness, change of, in achahatic ex- pansion, 93 fraction, 70, 81 Dust-free an, condensation in, 85 Ebullition, 349 Efficiency, conditions of maximum, 36 of a heat-engine, 2 of a perfect steam-engine, 90 of a i e versible heat-engine, 29 of Ranlane cycle, 99 -intio, 110 in a steam turbine, 210, 219 Elasticity of a fluid, 280 Encigv clue to suiface tension, 341- internal, 5, Ifi molecular, 2(>U, 357 of a gas 245, 357 gas-engino mixture, 251 of wbration in molecules, 301 Engine with sepaiate oigans, 90 workrng wi.thout expansion, 114 Entiopy 44 change of, in an in e\ersiblc opera- tion, 48 of a fluid, 75 ofwatei,334 sum of, in a system of bodies, tO temperature chart fm steam 118 temperatuie diagram for ic- fiigeiating cycle, 141 temperatuie diagiam of perfect steam engine, 91 -tempeiatiuc dmaiams, 40, 50 Equilibrium of diops, 345 Equipaitition of eneigj in tianslalion and rotation, 359 Etlnl chloride, 138 Eueken, A , 30(5 Evaporation, 05 Exhaust, 97 Expansion cylinder, omission of, in rc- Explosiuii, ladialion in, 2bO External \\uik in foimatmn of st( am, (>0 Extreme cold, pioduciion ot, 11)0 Feed pump, \\oik ^pcnt in 97 Fust la\\ ut thcimodv namics 5 Flames experiments \\ilh, 2t)2 Fluid cliaiiirtoiistiCKiiiatioiint 200,300 functions of tlio state uf, 77, 20(> Fluids properties ot, 50 Fon e, unit of 8 Four stroke eye It 1 , 220 Fiee eneig\ , 103 Friction, effects ol, in jets and (mimics, 209 Function ft, 1 02 Functions, Oilibs' theimodynamic, 103 of the state of a llmd, 77, 200 Gas, perfect, U, 290, 352 theimometcr, 11 Gas-constant, 238 Gas-engine, 225 mrxturo, energy of, 251 Gases, combustion of, 235 molecular energy of, 357 theoiy of, 351 piopertres of, ]3 specific heats of, 239, 357 Gauge-pressure, 05 General theruiodynamic relations, 206 INDEX 381 ~!ibbs, Willaid, 72, 103 ioodenough and Moshei, 1 J5 Joiiclio, W J , 218 Ji a mine-molecule, 237 LJimdluy J H , 32!) Heat account in a mal piocess, 120 dueel H -so of, to pioduce cold, 100, 104, 1(>7 ~ l.ittnt, 00, G8 muJnuuuil equivalent of, 10 of loi mat ion oi steam, (57 unit, of, ( ) Heal ih up, SI), 131 application to the theory of juts, 192 calculation of, 1UO tables, 110 Heat engine, clcliiulion of, 1 olluiency of, 2 lfi.it ituiup, 1, 1,!3, 1.50 I 1 ' ' i ot, 109 I I 103 Hi liiilioll/, It \on, 201 HCMU'S iciiLticm tuibine, 221 Hi use, W , 30I> ILinh uiuium, hcnclit <it, J 12 iinpcnlaiu.p ot, in atoiiin tin buus, 223 Ilollioin ind I kiimng, 2 r >3 llnpkinvm, I! , 240 2 IS, 251, 2 r > r >, 250, 2")7, 2 r iS, 2(>0 Jl\dii>< a ciLtii.il luiipii ilme ol, hi isol In i tnal i of 301 liquifm turn ill, 10 ( ) spriilit liuiL (if al \iiylott tompu.il nits 3l>0 Iii, ( Hi < I dl pnssnii on tlie mulling I'Dinl ol, I 17 ldi.il sli ,iiu Liiyino lollou mi; Cainots < \ i li , S,S liupuliil as, 2') r > 2 ( <) ImpiiUi I MI luiu , 220 IIK omploli i \paiiHion, 112, 11! link pi inli ill \ iin.ililis, 20,S Indu <iloi diayiain, 7 I ill i a i id spi 1 1 1 11 in ol awn, lilil liili'in.d i oinliiislion ongini', f>7 225 lull inal cm i\, r >, 1C ill a, fluid, (){) of a gas, 215, 357 Involution of c oolin^ elloc t, 277, Jld liiinusihlo ] n DCCHsos, 131 Iriotliininal c iiivos, 298 clawticity, 280 ovpantuon, 22, 78 of a poif ocL gas, 24 lines on tlio piOHnui'o volume chauiam, 78 Cm stoani 330 Joans, J Jl , 306, 303 Jonluu, G F , 145, 285 Jet-pump, use of, in leliigciation, 156 Jets, theoiy ot, 1U1 Joly, J , 242 Joule's au-ongine, 55 icversed, 158 equivalent, 10 law, 15 Joule Thomson cooling oflcct, 17, 127, 270, 290, 314, 318 Joule Thomson cooling cflect accoidmg to Callcndai's equation of state, 322 Joule Thomson coolmi; oiled, use of, by Lindo, 171 poiou9plugo\pciiment,74: Kelvin, Loul, 1(5, 39, 50, 117, 345 Koycs and Biownloe, 145 Kilo caloiy, 9 Kinetic theoiy of gases, 300, 351 Langon, 251 Latent hodt, 00, (58 Lmde, C, 171, 176, 181,27(5 Liqnof.n. lion ol gaacs, Lindc'u method, 170 Liquid lilms, tension in, 313 Ljungstiom tiiilnni 1 , 221 Low pimsuii dkatn, IHC of, in attain tuiliinis, 223 LuimiKi and Pim^sliiim, 215 JMall.ud and Le ChaUlic-i, 251 Mailin, II M , 2nS Matliot, 2M AIa\iinuin i UK H IK \ , i nndilions of ,!(> Ma\\\t H'sluiii llii unodj naniiL illations, 271 Mean tiio ]ialli, 351 - tin imal unit, ( ), f U Mn InuiKul i(|in\ tilt nt, 10 Milling poml, I'lU'i t of picssuie on, 117 Mi Uxlalili slali, 200 and oquililiiiuin o\p.nnifin, 205, 20S Mi\id nisis, picssiiu <if, 05 Mi\luio ol a liquid \\ilh ils Riiliaalid viijinui, 73, 70, 2SJ Mol, 237 IMoleuil.u eiiiigy ol a gas, 20 I , .557 lhcoi\ of glides, 351 \olocilus in LMHOS, 5 r >( woightH and \oliiinon, 235 Alolounlos, encigy of, 357 mimboi ol, in a cnliic cenli- mclio, 355 in a gi<iuini(j- molLculc, 355 Molhci, R, 121, 3K) cluut of total heat and ontiopy, 124 piG8surol25 chaits for subafcaucoa used in lefugciation, 14.5 382 INDEX Monatomic gas, 261, 357 Moss, H , 110 Nernst, W , 205 Newcomen, OS Newton's theory of transmission of sound, 29 i Nitiogen, isotheimals of, 301 separation of, 181 Nucleus, need of, in the formation of a diop, 344 Oblique coordinates, use of in the /</> chart, 145 Ols/sewski, 277 Oigans ot a heat engine, 96 Otto, 226 Oxygen, separation of, 181 Parsons' turbine, 216, 221 Paitial pressures, G5 Peake, A H , 320 Perfect ind imperfect gases, 306 ditteienlial, 270 engine, enter loti ot, 29 engine using regenerator, 50 gas, 14, 23S, 325 characteristic equation of, 291) expansion of, 22, 24 steam engine, 88 efficiency of, 90 eutiop} tempera- ture diagram of, 91 Petrol engine, 225 Phase, change of, 283 Phases of a substance, 103 Pier, 245 Planck's constant, 3b i formula ior energj'- of vibration in molecules, 362 Pol} atomic gases, specific heats, 364 molecules, 359 Portei, 3 IS Pound calory, 9 Pressure, of a gas, explanation of, on the molecuJai theory, 352 unit of, 8 -volume diagram, G Preston, 295 Radiation in explosions, 200 from dames, 262 Rankrue, 63, 70 cycle, 98 etticiency of, 102 for steam in any state, 104 reveisrbilrty of, 109 Rateau steam turbine, 220 Rayleigh, Loul, 294 Re action turbine, 221 Rectification, 181 Rectification, complete, 1SS ^ ' '74, 120 I. achme as a means oi warming, 1US ooLlIicicnlotnci- loimanco ol, 2 drlimlion of, 1 Re fiigtratron process, 133 Regenerative air -engine, 53 method ol producing ex- treme cold, 171 Regenerator, Strrlrng's, 52 Rcgnault, 14, 62, 241, 310 Re-heat factor, 218 Report ot Refrigeration Research Com- mrttee 145 Reiersibrhty, 20 conditions ol, 37 the criterion ot per foe Iron, 29 Rc\eisiblc engine efficiency of, 34 receiving licit at various temperatures, 42 heat engine, 27 reinitiating nu lime, lot Reynolds, Osbornc, 198 Rontgcn, 295 lUithertorcl, ,Srr E , 358 Saturated steam, 61, 02 61, !(>S ,570 relation ui pursuit' to tunpc laluu in, 33(> \apoui, 59 Saturation due to cm \alutc, 3-)5 ot an \\ Hli wati i-\ a |K nu, 66 Scale ot temperature theimnd\ ii.unie 12, 39 he heel, K , 360 Seay process, 103 Second la\v ot tluimud\ nauncs, 26 Mnt Ids, M C 1 , 361) Simple turbine, 215, 220 bpccihcation of stale otaiij fluid, 77, 2d7 Specific heat of \\alei, l>7 \anation ol, with tcuip- ciatiiu, 213 ' ' 1!) in C'allcndai's equation of state, 32-1 measurement oi value's ol, 241 of a gat), 17, 230, 357 of a gas, influence ol molecular vibration, 363 of gases on the molecular theorv 357 of hydrogen at ver^ low tcuipeiaturcs, 365 ratio of, 2 !, 275, 293, 3C7 Stage efficiency in tuibuics, 218 State, specification of, 77 States of aggregation, 59 INDEX 383 Steam, Callamlxi's lounulas foi, 327, Thomson, James, his ideal notheim.il, 330, 338 collcc Led fonnuliis 01, 338 cntical pic'MHuio of, 320 culical temper atuio of, SO cnliopy of, 37u Ruination o, imdoi constant pLessme, liO, 07 jets, supcisaliualion m, 2(13 piopeilics of, b2, 3o7 saluialod, 01, t>2, M piopoitioa of, 30S apecilic ho it ot, 378 Hiipeihoaled, 01 tal.lea, 3 10, 307 total heal of, 374 tuibiuo, poiunmaneo of, 222 Imhmes, compound, 216 mmplo, 215 typos ot, 220 volume of, 372 Mi un-i'iigiuo woikinLi, without coiu- piLisum, ') t Stilling, II , r >2 Slodola, 203 hut lion tt'inpoiadii o in a if. is engine, 2/37 Sndduv expansion, tltei I ul, S4 SulpliimuiH at ul, L iS hup< Konlin^;, Sf), 201 1 I^upi iluvitul \ iipoiu, f) ( ) tdl.il heat ol, 7 5 \\-Ui i, U'l Sll[n is Li ill lllnll, S I ol Mliaiu disi li mu lioni a uu//li , 201 Sin f.n <i It mion, 342 N\\ uiti, 212 Tables ot pi opi i tu s (it sti ,iiii, 3(>7 TLIIIJK latino, HI iili s ul, 10 of inviision ot iiHiliiiir t Hot t, 277 tin i muil\ iiiimii HI ak ol, 12, }<) I't in point in is in a u,an i n 2 r )5 Tension of liijind dim, 3 13 Thtimal unil used in Calluidars units, ') r l i ni i i l.i Thciitnodyiuiiuu ' T 'I ou ec lion ot Iho eulialH, 101 liom, 2d() (olIc('kul,2H7 )l toiupoialuio, 12, 3') suiliico, 2(58 HC10I1CO Of, 1 117, 304, 316 305, 317 Thomson, fcjn J J , 345 Thom-,011, W, Loul Kelvin, 1C, 3'), CO, 117,345 Thiollling caloiinioU'i, 128 cooling cAecfc of, 127, 27d, 29t> pioccsa, 74 Total heat, constancy of m a lluolllm piocoas, 74 ol a iluid, 70, 72 of watei, 334 Tnalnnno gas, 2(>1 molet'ulca, 350 Tiiplu point, 1 IS Tm hint s, tvpis ol, 220 Tuihn'mco, oilccti oi, 251) T\vo-sliol\.o cycln, 22(j Unit ol toicc, 8 lital, ') piestuuo, 8 woi k S Unit sisttil t-xpaiiHioii, 270 Van dui Waals 1 chai.u tonstic equation, 30(> Ihorii y tif toi itsponding 4lale4 311 V r .i[ioui < ompi tssiiin lolii^;* i at nig ninch- IIH, MS \.ipoui jiussuio ovoi a cin\L.d autfaco, 3l r > \ i loi il\ r ut nuan s(|uiiio, i r ) i, 3f>l) \ ilu ilion ul itdins in mul(< uli M, i()0 \ oliiiiK'ti IL spi i iii< h< ats, 23 f > Wal< i at sut m uf ion pussuir, piopulit'H ol, 3d 1 ) S|l( I ill! Ill III ol, ()7, I {"> supi i In al mi' ul, 3 IM int.ii iic'U mill i ni mp^ oi, ,n i \\ r uli i \ iipiiin n 1 1 ii>t i al inu mat hme, lf>f) Wall .lanus, ',S Wat I'M iiidii.itiu, 7 \\ r (iglil, vaiulion ol, with liititudi, 8 c ylnidi i , \Vi t nti urn, 7( Wilson, (' T R , .S, r >, 200, 207, 311, 3J' Wimpcin II U , 2f)0 \Vilkownki, (03 Woik dono by change of volume, (> done in adi.ihatio ovpaiiMinn, S(> uiul ot, H Working HnbHtance, 2 eyelo ol opuiatiuiiH ol, I f, S3'dncy, 313 CPHS, 137 Xouuoi, 03, 82 Zolly stoam turbmo, 220 CAMBRIDGE PRINTPD B\ J B PEACE, M A , AT THF UNIVERSITY PRESS \JMBRIDGE PRINTED BY J B PEACE, M A , AT THE UNIVERSITY PRESS